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Lemma 6.7. Let \( Q\left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4}}\right) \) be a quadrilateral with module 1, and let \( {s}_{1} \) and \( {s}_{2} \) denote the euclidean distances in \( Q \) between the sides \( \left( {{z}_{1},{z}_{2}}\right) ,\left( {{z}_{3},{z}_{4}}\right) \) and \( \left( {{z}_{2},{z}_{3}}\right) ,\left( {{z}_{4},{z}_{1}}\right) \), respectively. Then\n\n\[ \n{s}_{1}/{s}_{2} > {10}^{-3}\text{.} \n\]
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Proof. We may assume that among the arcs which join the sides \( \left( {{z}_{2},{z}_{3}}\right) \) and \( \left( {{z}_{4},{z}_{1}}\right) \) in \( Q \) there is a \( {\gamma }_{0} \) of length \( {s}_{2} \) . Let \( {z}_{0} \) be the point which divides \( {\gamma }_{0} \) into two parts of length \( {s}_{2}/2 \) . Set \( \rho \left( z\right) = 2/{s}_{2} \) if \( \left| {z - {z}_{0}}\right| \leq {s}_{2}/2,\rho \left( z\right) = 1/\left| {z - {z}_{0}}\right| \) if \( {s}_{2}/2 < \left| {z - {z}_{0}}\right| \leq {s}_{1} + {s}_{2}/2 \), and \( \rho \left( z\right) = 0 \) elsewhere. The area \( {m}_{\rho }\left( Q\right) \) of \( Q \) in this \( \rho \) -metric then satisfies the inequality\n\n\[ \n{m}_{\rho }\left( Q\right) \leq \pi \left( {1 + 2\log \left( {1 + 2{s}_{1}/{s}_{2}}\right) }\right) \n\]\n\nConsider next an arc \( \gamma \) joining the sides \( \left( {{z}_{1},{z}_{2}}\right) \) and \( \left( {{z}_{3},{z}_{4}}\right) \) in \( Q \) . For the \( \rho \) -length of \( \gamma \) we obtain a minorant if we integrate \( 1/x \) over a segment with endpoints \( {s}_{2}/2 \) and \( {s}_{2}/2 + {s}_{1} \) . Therefore,\n\n\[ \n{\int }_{\gamma }\rho \left( z\right) \left| {dz}\right| \geq \log \left( {1 + 2{s}_{1}/{s}_{2}}\right) \n\]\n\nSetting\n\n\[ \nF\left( x\right) = \frac{1 + 2\log \left( {1 + x}\right) }{{\left( \log \left( 1 + x\right) \right) }^{2}} \n\]\n\nwe thus we have by formula (1.7),\n\n\[ \n1 = M\left( {Q\left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4}}\right) }\right) \leq {\pi F}\left( {2{s}_{1}/{s}_{2}}\right) . \n\]\n\nFrom this we obtain, by interchanging the roles of \( {s}_{1} \) and \( {s}_{2} \),\n\n\[ \n\frac{{s}_{1}}{{s}_{2}} \geq \frac{2}{{F}^{-1}\left( {1/\pi }\right) } = \frac{2}{{e}^{\pi + {\left( {\pi }^{2} + \pi \right) }^{1/2}} - 1} > {10}^{-3}. \n\]
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Yes
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Theorem 6.6. A Jordan domain whose boundary satisfies the arc condition is a quasidisc.
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Proof. Let \( C \) be a Jordan curve which satisfies the arc condition and bounds the domains \( {A}_{1} \) and \( {A}_{2} \) . Choose four points \( {z}_{1},{z}_{2},{z}_{3},{z}_{4} \) on \( C \) such that \( {A}_{1}\left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4}}\right) \) is a quadrilateral with module 1 . We shall derive an upper bound for the module of the conjugate quadrilateral \( {A}_{2}\left( {{z}_{4},{z}_{3},{z}_{2},{z}_{1}}\right) \) . Let \( {s}_{1} \) denote the distance in \( {A}_{1} \) between the sides \( \left( {{z}_{1},{z}_{2}}\right) \) and \( \left( {{z}_{3},{z}_{4}}\right) \), and \( {d}_{1} \) the same distance measured in the plane. For the remaining sides \( \left( {{z}_{2},{z}_{3}}\right) \) and \( \left( {{z}_{4},{z}_{1}}\right) \) these distances are denoted by \( {s}_{2} \) and \( {d}_{2} \) . From Lemma 6.7 it follows that \[ {s}_{1} > {10}^{-3}{d}_{2} \] (6.9) Since \( C \) satisfies the arc condition, there is a constant \( k \geq 1 \) such that one of the sides \( \left( {{z}_{2},{z}_{3}}\right) ,\left( {{z}_{4},{z}_{1}}\right) \) lies inside a disc of diameter \( k{d}_{1} \) . From this we conclude that \[ {d}_{1} > \frac{{d}_{2}}{{10}^{3}{\pi k}} \] (6.10) For if not, one of the sides \( \left( {{z}_{2},{z}_{3}}\right) ,\left( {{z}_{4},{z}_{1}}\right) \) lies in a disc of diameter \( {10}^{-3}{d}_{2}/\pi \) . The other, which is at a distance \( {d}_{2} \) from this one, must lie outside this disc. It follows that the sides \( \left( {{z}_{1},{z}_{2}}\right) \) and \( \left( {{z}_{3},{z}_{4}}\right) \) can be joined in \( {A}_{1} \) by a circular arc of length \( \leq {10}^{-3}{d}_{2} \) . This is in contradiction with (6.9), and (6.10) follows. Since (6.10) is formulated in terms of the distances in the plane, it can be used to estimating the module of \( {A}_{2} \) . We first conclude as above the existence of a disc \( \left| {z - {z}_{0}}\right| < k{d}_{2}/2 \) which contains one of the sides \( \left( {{z}_{1},{z}_{2}}\right) ,\left( {{z}_{3},{z}_{4}}\right) \) . Let \[ r = k{d}_{2}/2 + {10}^{-3}{d}_{2}/{\pi k} \] and define \( \rho \left( z\right) = 1 \) if \( \left| {z - {z}_{0}}\right| < r \), and \( \rho \left( z\right) = 0 \) elsewhere. By (6.10), the \( \rho \) -length of an arc \( \gamma \) joining the first and the third side of \( {A}_{2} \) is \( \geq {10}^{-3}{d}_{2}/{\pi k} \) . Hence, \[ M\left( {{A}_{2}\left( {{z}_{4},{z}_{3},{z}_{2},{z}_{1}}\right) }\right) \leq {10}^{6}{\pi }^{3}{r}^{2}{k}^{2}/{d}_{2}^{2} = {10}^{6}\pi {\left( \pi {k}^{2}/2 + {10}^{-3}\right) }^{2}. \] By Lemma 6.6, \( C \) is a quasicircle, and the theorem is proved. We remark that \( C \) is a \( c\left( k\right) \) -quasicircle where \( c\left( k\right) \) depends only on the constant \( k \) in the arc condition.
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Yes
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Theorem 6.7. Let \( C \) be a \( K \) -quasicircle passing through \( \infty \), and \( {z}_{1},{z}_{2},{z}_{3} \) finite points of \( C \) such that \( {z}_{2} \) lies between \( {z}_{1} \) and \( {z}_{3} \) . Then\n\n\[ \left| {{z}_{1} - {z}_{2}}\right| + \left| {{z}_{2} - {z}_{3}}\right| \leq c\left( K\right) \left| {{z}_{1} - {z}_{3}}\right| . \]\n\n(6.11)
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Proof. Let \( f \) be a \( K \) -quasiconformal mapping of the plane which maps the real axis onto \( C \) such that \( f\left( \infty \right) = \infty \) . Denote \( {x}_{i} = {f}^{-1}\left( {z}_{i}\right), i = 1,2,3 \), and \( {C}_{1} = \left\{ {w\left| \right| w - {x}_{1}\left| = \right| {x}_{1} - {x}_{2} \mid }\right\} ,{C}_{2} = \left\{ {w\left| \right| w - {x}_{3}\left| = \right| {x}_{2} - {x}_{3} \mid }\right\} \) . Join \( {z}_{1} \) and \( {z}_{3} \) by a line segment \( L \), and denote by \( {a}_{1} \) and \( {a}_{2} \) the first points at which \( L \) meets \( f\left( {C}_{1}\right) \) and \( f\left( {C}_{2}\right) \) when one moves along \( L \) from \( {z}_{1} \) and from \( {z}_{3} \) . Then\n\n\[ \left| {{z}_{1} - {a}_{1}}\right| + \left| {{z}_{3} - {a}_{2}}\right| \leq \left| {{z}_{1} - {z}_{3}}\right| . \]\n\nBy Theorem 2.4,\n\n\[ \left| {{z}_{1} - {z}_{2}}\right| \leq c\left( K\right) \left| {{z}_{1} - {a}_{1}}\right| ,\;\left| {{z}_{2} - {z}_{3}}\right| \leq c\left( K\right) \left| {{z}_{3} - {a}_{2}}\right| . \]\n\nThese yield the desired inequality (6.11). It follows from the proof and our remark in 2.5 that (6.11) holds for \( c\left( K\right) = \lambda \left( K\right) \) .
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Yes
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Theorem 1.2 (Area Theorem). Let \( f \) be a univalent meromorphic function in the domain \( \{ z\left| \right| z \mid > 1\} \), with a power series expansion\n\n\[ f\left( z\right) = z + \mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{z}^{-n}. \]\n\nThen\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }n{\left| {b}_{n}\right| }^{2} \leq 1 \]\n\nThe inequality is sharp.
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Proof. Let \( {C}_{\rho } \) be the image of the circle \( \left| z\right| = \rho > 1 \) under \( f \) . The finite domain bounded by \( {C}_{\rho } \) has the area\n\n\[ {m}_{\rho } = \frac{i}{2}{\int }_{{C}_{\rho }}{wd}\bar{w} \]\n\nSubstituting \( w = f\left( z\right) \) and considering (1.14) we obtain\n\n\[ {m}_{\rho } = \pi {\rho }^{2} - \pi \mathop{\sum }\limits_{{n = 1}}^{\infty }n{\left| {b}_{n}\right| }^{2}{\rho }^{-{2n}}. \]\n\nAs an area, \( {m}_{\rho } > 0 \) and the result (1.15) follows as \( \rho \rightarrow 1 \) .
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Yes
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Theorem 1.3. Iff is a conformal mapping of a disc, then\n\n\[ \n\\begin{Vmatrix}{S}_{f}\\end{Vmatrix} \\leq 6\\text{.}\n\]\n\n(1.20)\n\nThe bound is sharp.
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Proof. By formula (1.12) it does not matter in which disc \( f \) is defined. We suppose that \( f \) is a conformal mapping of the unit disc \( D \) . Let us choose a point \( {z}_{0} \\in D \) and estimate \( \\left| {{S}_{f}\\left( {z}_{0}\\right) }\\right| \\eta {\\left( {z}_{0}\\right) }^{-2} = {\\left( 1 - {\\left| {z}_{0}\\right| }^{2}\\right) }^{2}\\left| {{S}_{f}\\left( {z}_{0}\\right) }\\right| \) . By (1.9), this expression is invariant under Möbius transformations. Hence, we may assume that \( {z}_{0} = 0 \) . Also, since \( f \) can be replaced by \( h \\circ f \), where \( h \) is an arbitrary Möbius transformation, there is no loss of generality in supposing that \( f \\in S \) . Let \( {a}_{n} \) denote the \( n \) th power series coefficient of \( f \) .\n\nThe function\n\n\[ z \\rightarrow 1/f\\left( {1/z}\\right) = z + \\mathop{\\sum }\\limits_{{n = 0}}^{\\infty }{b}_{n}{z}^{-n} \]\n\nsatisfies the conditions of the Area theorem. From \( {b}_{1} = {a}_{2}^{2} - {a}_{3} \) we thus conclude that \( \\left| {{a}_{2}^{2} - {a}_{3}}\\right| \\leq 1 \) . On the other hand, \( {S}_{f}\\left( 0\\right) = 6\\left( {{a}_{3} - {a}_{2}^{2}}\\right) \) . Consequently, \( \\left| {{S}_{f}\\left( 0\\right) }\\right| \\leq 6 \), and (1.20) follows.\n\nFor the Koebe functions \( f \) the coefficient \( {b}_{1} \) of \( z \\rightarrow 1/f\\left( {1/z}\\right) \) is of absolute value 1. Hence, for the Koebe functions \( \\left| {{a}_{2}^{2} - {a}_{3}}\\right| = 1 \), and equality holds in (1.20). More generally, in \( D \) equality holds in (1.20) for all functions \( h \\circ f \\circ g \) , where \( g \) is a conformal self-mapping of \( D, f \) a Koebe function, and \( h \) an arbitrary Möbius transformation. In the upper half-plane, \( z \\rightarrow f\\left( z\\right) = {z}^{2} \) is a simple example of a univalent function for which \( \\begin{Vmatrix}{S}_{f}\\end{Vmatrix} = 6 \) .\n\nThe estimate (1.20) was proved by Kraus [1] in 1932. His paper was forgotten and rediscovered only in the late sixties. Meanwhile, (1.20) was attributed to Nehari ([1]) who proved it in 1949.
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Yes
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Theorem 2.1. If \( A \) is Möbius equivalent to a convex domain, then\n\n\[ \delta \left( A\right) \leq 2\text{.} \]\n\nEquality holds if \( A \) is the image of a parallel strip under a Möbius transformation.
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Proof. We may assume that \( A \) itself is convex. Let \( f \) be an arbitrary conformal mapping of \( D \) onto \( A \) . In view of (2.2), inequality (2.9) follows if we prove that \( \left| {{S}_{f}\left( 0\right) }\right| \leq 2 \) . Since we may replace \( f \) by the function \( z \rightarrow {cf}\left( {z{e}^{i\varphi }}\right) \) for \( c \) complex and \( \varphi \) real, there is no loss of generality in assuming that \( {S}_{f}\left( 0\right) \geq 0 \) and that \( {f}^{\prime }\left( 0\right) = 1 \) .\n\nFrom (2.8) we obtain by direct computation\n\n\[ {S}_{f}\left( 0\right) = {\int }_{0}^{2\pi }{e}^{-{2i\theta }}{d\psi }\left( \theta \right) - \frac{1}{2}{\left( {\int }_{0}^{2\pi }{e}^{-{i\theta }}d\psi \left( \theta \right) \right) }^{2}. \]\n\n(2.10)\n\nSince \( {S}_{f}\left( 0\right) \) is real and \( {d\psi }\left( \theta \right) \geq 0 \), it follows that\n\n\[ {S}_{f}\left( 0\right) = {\int }_{0}^{2\pi }\cos {2\theta d\psi }\left( \theta \right) - \frac{1}{2}{\left( {\int }_{0}^{2\pi }\cos \theta d\psi \left( \theta \right) \right) }^{2} + \frac{1}{2}{\left( {\int }_{0}^{2\pi }\sin \theta d\psi \left( \theta \right) \right) }^{2} \]\n\n\[ \leq {\int }_{0}^{2\pi }\cos {2\theta d\psi }\left( \theta \right) - \frac{1}{2}{\left( {\int }_{0}^{2\pi }\cos \theta d\psi \left( \theta \right) \right) }^{2} + {\int }_{0}^{2\pi }{\sin }^{2}{\theta d\psi }\left( \theta \right) \]\n\n(2.11)\n\n\[ = {\int }_{0}^{2\pi }{\cos }^{2}{\theta d\psi }\left( \theta \right) - \frac{1}{2}{\left( {\int }_{0}^{2\pi }\cos \theta d\psi \left( \theta \right) \right) }^{2} \leq {\int }_{0}^{2\pi }{\cos }^{2}{\theta d\psi }\left( \theta \right) \leq 2. \]\n\nBecause \( {S}_{f}\left( 0\right) \geq 0 \), we have proved (2.9).\n\nEquality holds if\n\n\[ {\int }_{0}^{2\pi }{\cos }^{2}{\theta d\psi }\left( \theta \right) = 2,\;{\int }_{0}^{2\pi }\cos {\theta d\psi }\left( \theta \right) = 0. \]\n\nThese conditions are fulfilled if \( \psi \) has a jump +1 at the points 0 and \( \pi \) and is constant on the intervals \( \left( {0,\pi }\right) \) and \( \left( {\pi ,{2\pi }}\right) \) . Then \( {S}_{f}\left( 0\right) = 2 \), and it follows from (2.8) that \( {f}^{\prime }\left( z\right) = {\left( 1 - {z}^{2}\right) }^{-1} \) . We conclude that the image of \( D \) is a parallel strip.
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Yes
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Theorem 2.2. Let \( A \) be Möbius equivalent to a domain with boundary rotation \( \leq {k\pi } \) . If \( k \leq 4 \), then \[ \delta \left( A\right) \leq \frac{{2k} + 4}{6 - k}. \]
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The main lines of the proof are the same as those in Theorem 2.1. After similar initial remarks we start from (2.10), assuming this time that \( {S}_{f}\left( 0\right) < 0 \) . In the first line of (2.11) we now ignore the third integral and conclude that \[ \left| {{S}_{f}\left( 0\right) }\right| \leq \frac{1}{2}{\left( {\int }_{0}^{2\pi }\cos \theta d\psi \left( \theta \right) \right) }^{2} - {\int }_{0}^{2\pi }\cos {2\theta d\psi }\left( \theta \right) . \] With attention paid to (2.4) and (2.5), the estimate (2.12) follows from this after some computation; for the details we refer to Lehto and Tammi [1], p. 255.
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No
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Theorem 2.3. Let \( A \) and \( {A}^{\prime } \) be domains conformally equivalent to a disc and \( f : A \rightarrow {A}^{\prime } \) a conformal mapping. Then\n\n\[ \n{\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} \leq \delta \left( A\right) + \delta \left( {A}^{\prime }\right) \n\]\n\n(2.13)\n\nThe estimate is sharp for any given pair of domains \( A \) and \( {A}^{\prime } \) .
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Proof. Let \( h \) be a conformal mapping of the unit disc \( D \) onto \( A \) . From \( f = \left( {f \circ h}\right) \circ {h}^{-1} \) we conclude that \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} = {\begin{Vmatrix}{S}_{f \circ h} - {S}_{h}\end{Vmatrix}}_{D} \) . Since\n\n\[ \n{\begin{Vmatrix}{S}_{h}\end{Vmatrix}}_{D} = \delta \left( A\right) ,\;{\begin{Vmatrix}{S}_{f \circ h}\end{Vmatrix}}_{D} = \delta \left( {A}^{\prime }\right) ,\n\]\n\nthe triangle inequality yields (2.13).\n\nIn order to verify that the estimate (2.13) cannot be improved, we consider conformal mappings \( {h}_{1} : D \rightarrow A,{h}_{2} : D \rightarrow {A}^{\prime } \) . Given an \( \varepsilon > 0 \), we choose \( {h}_{1} \) and \( {h}_{2} \) such that\n\n\[ \n\left| {{S}_{{h}_{i}}\left( 0\right) }\right| > {\begin{Vmatrix}{S}_{{h}_{i}}\end{Vmatrix}}_{D} - \varepsilon ,\;i = 1,2.\n\]\n\n(2.14)\n\nThis is possible, because \( \left| {{S}_{f}\left( z\right) }\right| \eta {\left( z\right) }^{-2} \) is invariant under Möbius transformations (cf. the reasoning in 2.2).\n\nLet \( g \) be the rotation \( z \rightarrow {e}^{i\theta }z \) . Then \( f = {h}_{2} \circ g \circ {h}_{1}^{-1} \) maps \( A \) conformally onto \( {A}^{\prime } \), and\n\n\[ \n{\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} = {\begin{Vmatrix}{S}_{{h}_{2} \circ g} - {S}_{{h}_{1}}\end{Vmatrix}}_{D} \n\]\n\nNow\n\n\[ \n{\begin{Vmatrix}{S}_{{h}_{2} \circ g} - {S}_{{h}_{1}}\end{Vmatrix}}_{D} \geq \left| {{S}_{{h}_{2} \circ g}\left( 0\right) - {S}_{{h}_{1}}\left( 0\right) }\right| = \left| {{S}_{{h}_{2}}\left( 0\right) {e}^{2i\theta } - {S}_{{h}_{1}}\left( 0\right) }\right| .\n\]\n\nFor a suitable \( \theta \) we obtain from this and (2.14),\n\n\[ \n{\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} \geq \left| {{S}_{{h}_{2}}\left( 0\right) }\right| + \left| {{S}_{{h}_{1}}\left( 0\right) }\right| > \delta \left( {A}^{\prime }\right) + \delta \left( A\right) - {2\varepsilon }.\n\]\n\nConsequently, (2.13) is sharp.
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Yes
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Theorem 2.4. Let \( A \) be a domain conformally equivalent to a disc. Then\n\n\[ \n{\sigma }_{0}\left( A\right) = \delta \left( A\right) + 6 \n\]
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Proof. We write the definition of \( {\sigma }_{0}\left( A\right) \) in the form\n\n\[ \n{\sigma }_{0}\left( A\right) = \mathop{\sup }\limits_{{A}^{\prime }}\left\{ {{\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} \mid f : A \rightarrow {A}^{\prime }\text{ conformal }}\right\} .\n\]\n\nThen it follows from Theorem 2.3 that\n\n\[ \n{\sigma }_{0}\left( A\right) = \delta \left( A\right) + \mathop{\sup }\limits_{{A}^{\prime }}\delta \left( {A}^{\prime }\right) \n\]\n\nHence, we obtain (2.15) from Theorem 1.3.
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Yes
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Let \( \mu \) be a measurable function in the plane with bounded support and \( \parallel \mu {\parallel }_{\infty } < 1 \) . Let \( z \rightarrow f\left( {z, w}\right) \) be the quasiconformal mapping of the plane with complex dilatation \( {w\mu } \) and with the property \( \lim \left( {f\left( {z, w}\right) - z}\right) = 0 \) as \( z \rightarrow \infty \) . Then, for every fixed \( z \neq \infty \), the function \( w \rightarrow f\left( {z, w}\right) \) is holomorphic in the disc \( \left| w\right| < 1/\parallel \mu {\parallel }_{\infty } \) .
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Proof. By Theorem I.4.3,\n\n\[ f\left( z\right) = f\left( {z,1}\right) = z + \mathop{\sum }\limits_{{i = 1}}^{\infty }T{\varphi }_{i}\left( \mu \right) \left( z\right) ,\]\n\nwhere we now write \( {\varphi }_{i}\left( \mu \right) \) instead of \( {\varphi }_{i} \) to accentuate the dependence of \( {\varphi }_{i} \) on \( \mu \) . From the definition of the functions \( {\varphi }_{i}\left( \mu \right) \) it follows that\n\n\[ {\varphi }_{i}\left( {w\mu }\right) = {w}^{i}{\varphi }_{i}\left( \mu \right) \]\n\nHence,\n\n\[ f\left( {z, w}\right) = z + \mathop{\sum }\limits_{{i = 1}}^{\infty }T{\varphi }_{i}\left( \mu \right) \left( z\right) {w}^{i}. \]\n\n(3.1)\n\nFrom formula (4.15) in I.4.4 we see that \( \sum T{\varphi }_{i}\left( \mu \right) \left( z\right) \) converges uniformly whenever \( \parallel \mu {\parallel }_{\infty } < 1 \) . It follows that the power series (3.1) converges if \( \left| w\right| \parallel \mu {\parallel }_{\infty } < 1 \) . Consequently, \( w \rightarrow f\left( {z, w}\right) \) is analytic in the disc \( \left| w\right| < 1/\parallel \mu {\parallel }_{\infty } \) .\n\nOutside the support of \( \mu \), the function \( z \rightarrow f\left( {z, w}\right) \) is a conformal mapping. Also, each function \( z \rightarrow T{\varphi }_{i}\left( \mu \right) \left( z\right) \) is holomorphic, and \( T{\varphi }_{i}\left( \mu \right) \left( z\right) \) is no longer a singular integral. Therefore, we can differentiate in (3.1) with respect to \( z \) term by term, without affecting the convergence of the series. If prime denotes differentiation with respect to \( z \), we obtain\n\n\[ {f}^{\prime }\left( {z, w}\right) = 1 + \mathop{\sum }\limits_{{i = 1}}^{\infty }{\left( T{\varphi }_{i}\left( \mu \right) \right) }^{\prime }\left( z\right) {w}^{i}, \]\n\nand similarly for higher derivatives. It follows that all derivatives of \( z \rightarrow f\left( {z, w}\right) \) depend holomorphically on \( w \) in the disc \( \left| w\right| < 1/\parallel \mu {\parallel }_{\infty } \) .
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Yes
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Let \( \mu \) be a measurable function in the plane which vanishes in the upper half-plane and for which \( \parallel \mu {\parallel }_{\infty } < 1 \) . Let \( {f}_{w\mu } \) be the quasiconformal mapping of the plane with complex dilatation \( {w\mu } \) which keeps the points \( 0,1,\infty \) fixed. Then the function \( w \rightarrow {f}_{w\mu }\left( z\right) \) is holomorphic in \( \left| w\right| < 1/\parallel \mu {\parallel }_{\infty } \) for every finite \( z \) .
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Let \( g \) be the Möbius transformation which maps the points \( 0,1,\infty \) on the points \( - 1, i,1 \), respectively, and \( {\widetilde{f}}_{wv} \) a quasiconformal mapping of the plane whose complex dilatation \( {wv} \) agrees with that of \( {f}_{w\mu } \circ {g}^{-1} \) . Then\n\n\[ \mu \left( z\right) = v\left( {g\left( z\right) }\right) \overline{{g}^{\prime }\left( z\right) }/{g}^{\prime }\left( z\right) . \]\n\nFurther, let \( {\widetilde{f}}_{wv} \) satisfy the normalization condition \( {\widetilde{f}}_{wv}\left( \zeta \right) - \zeta \rightarrow 0 \) as \( \zeta \rightarrow \infty \) . By Theorem 3.1, \( w \rightarrow {\widetilde{f}}_{wv}\left( {g\left( z\right) }\right) \) is analytic in \( \left| w\right| < 1/\parallel v{\parallel }_{\infty } = 1/\parallel \mu {\parallel }_{\infty } \) . Set \( {a}_{1} = {\widetilde{f}}_{wv}\left( {-1}\right) ,{a}_{2} = {\widetilde{f}}_{wv}\left( i\right) ,{a}_{3} = {\widetilde{f}}_{wv}\left( 1\right) \), and\n\n\[ {h}_{w}\left( \zeta \right) = \frac{{a}_{2} - {a}_{3}}{{a}_{2} - {a}_{1}}\frac{\zeta - {a}_{1}}{\zeta - {a}_{3}}. \]\n\nThen \( {h}_{w} \circ {\widetilde{f}}_{wv} \circ g \) has the complex dilatation \( {w\mu } \), and it fixes \( 0,1,\infty \) . Consequently,\n\n\[ {f}_{w\mu }\left( z\right) = {h}_{w}\left( {{\widetilde{f}}_{wv}\left( {g\left( z\right) }\right) }\right) \]\n\nBy applying Theorem 3.1 again, we conclude that \( {h}_{w}\left( \zeta \right) \) depends analytically on \( w \) . It follows that \( w \rightarrow {f}_{w\mu }\left( z\right) \) is holomorphic in the disc claimed.
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Yes
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Theorem 3.2. Let \( f \) be a quasiconformal mapping of the plane which has the complex dilatation \( \mu \) and which is conformal in a simply connected domain \( A \) with at least two boundary points. Then\n\n\[ \n{\begin{Vmatrix}{S}_{f \mid A}\end{Vmatrix}}_{A} \leq {\sigma }_{0}\left( A\right) \parallel \mu {\parallel }_{\infty }.\n\]
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Proof. If \( g \) is a Möbius transformation, we can replace \( f \) by \( f \circ g \) without changing the norms of either the Schwarzian derivative or the complex dilatation. Also, \( {\sigma }_{0}\left( A\right) = {\sigma }_{0}\left( {{g}^{-1}\left( A\right) }\right) \) . We may therefore assume that \( \infty \in A \) . Then \( \mu \) has bounded support.\n\nLet \( w \) be a complex number with \( \left| w\right| < 1 \) . We consider for a moment the unique quasiconformal mapping \( z \rightarrow f\left( {z, w/\parallel \mu {\parallel }_{\infty }}\right) \) of the plane which has the complex dilatation \( {w\mu }/\parallel \mu {\parallel }_{\infty } \) and the property \( f\left( {z, w/\parallel \mu {\parallel }_{\infty }}\right) - z \rightarrow 0 \) as \( z \rightarrow \infty \) . (We may assume that \( \parallel \mu {\parallel }_{\infty } > 0 \) .) By Theorem 3.1, the derivatives of the analytic function \( z \rightarrow \left( {f \mid A}\right) \left( {z, w/\parallel \mu {\parallel }_{\infty }}\right) \) with respect to \( z \) depend analytically on \( w \) in the unit disc, at every finite point \( z \) of \( A \) .\n\nKeeping \( z \) fixed, we define the function\n\n\[ \nw \rightarrow \psi \left( w\right) = {S}_{f\left( {\cdot, w/\parallel \mu \parallel \infty }\right) }\left( z\right) {\eta }_{A}{\left( z\right) }^{-2}.\n\]\n\nSince \( {S}_{f} \) is a rational function of the first three derivatives of \( f \mid A \), we conclude that \( \psi \) is analytic in the unit disc \( \left| w\right| < 1 \) . Furthermore, the function \( \psi \) is bounded: \( \left| {\psi \left( w\right) }\right| \leq {\sigma }_{0}\left( A\right) \) . From the fact that \( z \rightarrow f\left( {z,0}\right) \) is the identity mapping it follows that \( \psi \left( 0\right) = 0 \) . We can therefore apply Schwarz’s lemma to \( \psi \) and get\n\n\[ \n\left| {\psi \left( w\right) }\right| \leq {\sigma }_{0}\left( A\right) \left| w\right|\n\]\n\nSetting \( w = \parallel \mu {\parallel }_{\infty } \), we get back, modulo a Möbius transformation, the function \( z \rightarrow f\left( z\right) \) we started with, and (3.2) follows.
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Yes
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Theorem 3.3. If \( A \) is a \( K \) -quasidisc, then\n\n\[ \delta \left( A\right) \leq 6\frac{{K}^{2} - 1}{{K}^{2} + 1} \]
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Proof. By Lemma I.6.2, the domain \( A \) is the image of the upper half-plane \( H \) under a \( {K}^{2} \) -quasiconformal mapping \( f \) of the plane which is conformal in \( H \) . By Theorem 3.2,\n\n\[ {\begin{Vmatrix}{S}_{f \mid H}\end{Vmatrix}}_{H} \leq 6\frac{{K}^{2} - 1}{{K}^{2} + 1}. \]\n\nOn the other hand, \( {\begin{Vmatrix}{S}_{f \mid H}\end{Vmatrix}}_{H} = \delta \left( A\right) \).
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Yes
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Theorem 3.4. Let \( f \in {\sum }_{k} \) and \( k < {k}_{0} < 1 \) . As \( k \rightarrow 0 \) ,\n\n\[ f\left( z\right) = z - \frac{1}{\pi }{\iint }_{D}\frac{\mu \left( \zeta \right) }{\zeta - z}{d\xi d\eta } + O\left( {k}^{2}\right) \]\n\nin the whole plane. Here \( \left| {O\left( {k}^{2}\right) }\right| \leq c{k}^{2} \), the constant \( c \) depending only on \( {k}_{0} \).
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Proof. If \( p > 2 \) and \( {k}_{0}\parallel H{\parallel }_{p} < 1 \), we see from formula (4.15) in I.4.4 that\n\n\[ \mathop{\sum }\limits_{{i = 2}}^{\infty }\left| {T{\varphi }_{i}\left( z\right) }\right| \leq {c}_{p}^{\prime }\mathop{\sum }\limits_{{i = 2}}^{\infty }{\left( k\parallel H{\parallel }_{p}\right) }^{i} \leq c{k}^{2}. \]\n\nHence, Theorem 3.4 follows from Theorem I.4.3.
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No
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Corollary 3.2. The functions \( f \in {\sum }_{k} \) satisfy the asymptotic inequality\n\n\[ \left| {f\left( z\right) - z}\right| \leq \frac{k}{\pi }{\iint }_{D}\frac{d\xi d\eta }{\left| \zeta - z\right| } + c{k}^{2}. \]
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If\n\n\[ \mu \left( \zeta \right) = k{e}^{i\theta }\frac{\zeta - z}{\left| \zeta - z\right| }\;\text{ a.e.,} \]\n\nthen\n\n\[ \left| {f\left( z\right) - z}\right| = \frac{k}{\pi }{\iint }_{D}\frac{d\xi d\eta }{\left| \zeta - z\right| } + O\left( {k}^{2}\right) . \]
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Yes
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Theorem 3.5. In the class \( {\sum }_{k} \), \[ \left| {b}_{n}\right| \leq \frac{2k}{n + 1} + c{k}^{2},\;n = 1,2,\ldots , \] with \( c \leq {n}^{-1/2}{\left( 1 - k\right) }^{-1} \). If \[ {f}_{n}\left( z\right) = \left\{ \begin{array}{ll} {\left( {z}^{\left( {n + 1}\right) /2} + k{z}^{-\left( {n + 1}\right) /2}\right) }^{2/\left( {n + 1}\right) } & \text{ if }\left| z\right| > 1, \\ {\left( {z}^{\left( {n + 1}\right) /2} + k{\bar{z}}^{\left( {n + 1}\right) /2}\right) }^{2/\left( {n + 1}\right) } & \text{ if }\left| z\right| \leq 1, \end{array}\right. \] then \( {f}_{n} \in {\sum }_{k} \) and \( {b}_{n} = {2k}/\left( {n + 1}\right) \) .
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Proof. We have \[ T{\varphi }_{i}\left( z\right) = - \frac{1}{\pi }{\iint }_{D}\frac{{\varphi }_{i}\left( \zeta \right) }{\zeta - z}{d\xi d\eta } = \frac{1}{\pi }\mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{\iint }_{D}{\varphi }_{i}\left( \zeta \right) {\zeta }^{n - 1}{d\xi d\eta }}\right) {z}^{-n} \] for \( \left| z\right| > 1 \). Hence, \[ {b}_{n} = \frac{1}{\pi }\mathop{\sum }\limits_{{i = 1}}^{\infty }{\iint }_{D}{\varphi }_{i}\left( \zeta \right) {\zeta }^{n - 1}{d\xi d\eta } \] Schwarz’s inequality and the estimate (4.13) in I.4.4 for \( p = 2 \) (in which case \( \parallel H{\parallel }_{p} = 1 \) ) yield \[ \left| {{\iint }_{D}{\varphi }_{i}\left( \zeta \right) {\zeta }^{n - 1}{d\xi d\eta }}\right| \leq {\pi }^{1/2}{n}^{-1/2}{\begin{Vmatrix}{\varphi }_{i}\end{Vmatrix}}_{2} \leq \pi {n}^{-1/2}{k}^{i}. \] Consequently, we have the asymptotic representation \[ {b}_{n} = \frac{1}{\pi }{\iint }_{D}\mu \left( \zeta \right) {\zeta }^{n - 1}{d\xi d\eta } + O\left( {k}^{2}\right) ,\;n = 1,2,\ldots , \] the remainder term being \( \leq {n}^{-1/2}{k}^{2}/\left( {1 - k}\right) \) in absolute value. From this (3.9) follows. We conclude by easy computation that \( {b}_{n} = {2k}/\left( {n + 1}\right) + O\left( {k}^{2}\right) \) if \[ \mu \left( \zeta \right) = k{\left( \bar{\zeta }/\zeta \right) }^{\left( {n - 1}\right) /2}\;\text{ a.e. } \] Direct verification shows that the function \( {f}_{n} \) defined by (3.10) has this complex dilatation. We also see that for \( {f}_{n} \), the coefficient \( {b}_{n} = {2k}/\left( {n + 1}\right) \) .
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Yes
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Theorem 3.6. Let \( \Phi \) be an analytic functional on \( \sum \) which vanishes for the identity mapping. Then \( M\left( k\right) /k \) is non-decreasing on the interval \( \left( {0,1}\right) \) .
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Proof. Fix \( k \) and \( {k}^{\prime },0 < k < {k}^{\prime } < 1 \), and choose an arbitrary mapping \( {f}_{0} \in {\sum }_{k} \) . Let \( \mu \) be the complex dilatation of some extension of \( {f}_{0},\parallel \mu {\parallel }_{\infty } \leq k \) . Consider the mappings \( f \in {\sum }_{{k}^{\prime }} \) which have the complex dilatation \( {w\mu } \) with \( \left| w\right| < {k}^{\prime }/k \) . By Theorem 3.1, \( \Phi \left( f\right) \) depends holomorphically on \( w \) in the disc \( \left| w\right| < {k}^{\prime }/k \) . If \( w = 0 \), then \( f \) is the identity mapping, so that \( \Phi \left( f\right) \) vanishes at \( w = 0 \) . Therefore, by Schwarz's lemma\n\n\[ \left| {\Phi \left( f\right) }\right| \leq \frac{k}{{k}^{\prime }}M\left( {k}^{\prime }\right) \left| w\right| \]\n\nFor \( w = 1 \) we have \( f = {f}_{0} \) . Since \( {f}_{0} \) is an arbitrary element of \( {\sum }_{k} \), we get the desired inequality \( M\left( k\right) \leq {kM}\left( {k}^{\prime }\right) /{k}^{\prime } \) .
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Yes
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Corollary 3.3 (Majorant Principle). If \( \Phi \) is an analytic functional on \( \sum \) which vanishes for the identity mapping, then\n\n\[ \mathop{\max }\limits_{{f \in {\sum }_{k}}}\left| {\Phi \left( f\right) }\right| \leq k\mathop{\max }\limits_{{f \in \sum }}\left| {\Phi \left( f\right) }\right| \]\n\n(3.13)\n\nIf equality holds for one value \( k \in \left( {0,1}\right) \), then it holds for all values of \( k \) .
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Proof. Inequality (3.13) follows immediately from Theorem 3.6 if we let \( k \rightarrow 1 \) .\n\nSuppose that equality holds in (3.13) for some value \( k,0 < k < 1 \) . Let \( {f}_{k} \) be extremal in this \( {\sum }_{k} \) and \( \mu \) the complex dilatation of its extension. For functions \( f \) with complex dilatation \( {w\mu },\left| w\right| < 1/k \), Schwarz’s lemma gives \( \left| {\Phi \left( f\right) }\right| \leq {kM}\left( 1\right) \left| w\right| \), where \( M\left( 1\right) = \max \left| {\Phi \left( f\right) }\right| \) over \( \sum \) . But now equality holds for \( w = 1 \) . It follows that\n\n\[ \left| {\Phi \left( f\right) }\right| = {kM}\left( 1\right) \left| w\right| \]\n\n(3.14)\n\nin the whole disc \( \left| w\right| < 1/k \) . If \( {k}^{\prime } \in \lbrack 0,1) \) is arbitrarily given, then for \( w = {k}^{\prime }/k \) the function \( f \) is in \( {\sum }_{{k}^{\prime }} \) . Combining (3.13) and (3.14) we deduce that \( f \) is extremal in \( {\sum }_{{k}^{\prime }} \) . In other words, if equality holds in (3.13) for one value \( k \in \left( {0,1}\right) \), then it holds for all values of \( k \), and if \( \mu \) is an extremal complex dilatation, then all dilatations \( {w\mu },\left| w\right| < 1/\parallel \mu {\parallel }_{\infty } \), are extremal.
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Yes
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Theorem 3.7. In the class \( {\sum }_{k} \) , \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }n{\left| {b}_{n}\right| }^{2} \leq {k}^{2} \] (3.16) The estimate is sharp.
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Proof. Given an arbitrary function \( f \in {\sum }_{k} \) with the coefficients \( {b}_{n} \), we set \( {\lambda }_{n} = {\left| {b}_{n}\right| }^{2}/{b}_{n}^{2} \) if \( {b}_{n} \neq 0 \) ; otherwise \( {\lambda }_{n} = 1 \) . Let \( \mu \) be the complex dilatation of the extended \( f \), and \( {b}_{n}\left( w\right) \) the coefficients of the function \( z \rightarrow f\left( {z, w/k}\right) \) with \( \sum \) -normalization and complex dilatation \( {w\mu }/k \) . For an arbitrary positive integer \( N \), we write \[ \psi \left( w\right) = \mathop{\sum }\limits_{{n = 1}}^{N}n{\lambda }_{n}{b}_{n}{\left( w\right) }^{2} \] By Theorem 3.1, \( \psi \) is holomorphic in the unit disc. The Area theorem gives the estimate \( \left| {\psi \left( w\right) }\right| \leq 1 \) . Since \( {b}_{n}\left( 0\right) = 0 \), the function \( \psi \) has a zero of order \( \geq 2 \) at the origin. Schwarz’s lemma therefore yields the improved estimate \[ \left| {\psi \left( w\right) }\right| \leq {\left| w\right| }^{2} \] (3.17) If we set \( w = k \), we get back the function \( f \) with which we started. Hence \[ \mathop{\sum }\limits_{{n = 1}}^{N}n{\lambda }_{n}{b}_{n}^{2} = \mathop{\sum }\limits_{{n = 1}}^{N}n{\left| {b}_{n}\right| }^{2} \leq {k}^{2}. \] The desired inequality (3.16) follows as \( N \rightarrow \infty \) . Equality holds in (3.16) if \( f\left( z\right) = z + k{e}^{i\theta }/z \) in \( \left| z\right| > 1 \) and \( f\left( z\right) = z + k{e}^{i\theta }\bar{z} \) in \( \left| z\right| \leq 1 \) . A relatively simple argument shows that there are no other ex-tremals (Lehto [1]). The result (3.16) is due to Kühnau [2] and Lehto [1].
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Yes
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Theorem 4.2. A Schwarzian domain is a quasidisc.
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Proof. Let \( A \) be an \( a \) -Schwarzian domain. Then \( A \) is trivially \( {a}^{\prime } \) -Schwarzian for \( {a}^{\prime } \leq a \) . We may suppose, therefore, that \( a \leq 2 \) . (In III. 5 we shall show that, in fact, no domain \( A \) is \( a \) -Schwarzian for \( a > 2 \), but here this result is not needed.) Nor is there any loss of generality in assuming that \( \infty \) does not lie in \( A \) .\n\nWe shall show that \( A \) is linearly locally connected with constant\n\n\[ c = 1 + {16}/a\text{.}\]\n\n(4.15)\n\nThe theorem then follows from Theorems I.6.5 and I.6.6. More precisely, we conclude that an \( a \) -Schwarzian domain is a \( K\left( a\right) \) -quasidisc, where the constant \( K\left( a\right) \) depends only on \( a \) .\n\nThe proof is indirect. Suppose that \( A \) is not linearly locally connected with the constant \( c = 1 + {16}/a \) . By Lemma 4.1, there are points \( {z}_{1},{z}_{2} \) in \( A \) and \( {w}_{1} \) , \( {w}_{2} \) outside \( A \), such that (4.10) holds. Clearly \( h\left( {z}_{1}\right) \neq h\left( {z}_{2}\right) \) .\n\nDefine\n\n\[ f\left( z\right) = {e}^{{bh}\left( z\right) },\;b = \frac{2\pi i}{h\left( {z}_{1}\right) - h\left( {z}_{2}\right) }.\]\n\nThen \( f\left( {z}_{1}\right) /f\left( {z}_{2}\right) = 1 \), so that \( f \) is not univalent.\n\nFrom \( {S}_{f} = - {b}^{2}{h}^{\prime 2}/2 + {S}_{h} \) we get by an easy computation\n\n\[ {S}_{f}\left( z\right) = \frac{1 - {b}^{2}}{2}{\left( \frac{{w}_{1} - {w}_{2}}{\left( {z - {w}_{1}}\right) \left( {z - {w}_{2}}\right) }\right) }^{2}.\]\n\nIf \( \eta \) denotes the Poincaré density of \( A \), then formula (1.5) in I.1.1 yields the estimate\n\n\[ \left| {{S}_{f}\left( z\right) }\right| \eta {\left( z\right) }^{-2} \leq 8\left| {{b}^{2} - 1}\right| .\]\n\n(4.16)\n\nSince \( a \leq 2 \), we see from (4.15) that \( c - 1 \geq 8 \) . Then \( \left| {h\left( {z}_{1}\right) - h\left( {z}_{2}\right) }\right| \geq \) \( {2\pi } - 1/2 \), and so\n\n\[ \left| {b - 1}\right| \leq \frac{4}{\left( {c - 1}\right) \left( {{2\pi } - 1/2}\right) } < \frac{4}{5\left( {c - 1}\right) } \leq \frac{1}{10}.\]\n\nIt follows that\n\n\[ \left| {{b}^{2} - 1}\right| \leq \frac{21}{10} \cdot \frac{4}{5\left( {c - 1}\right) } < \frac{a}{8}.\]\n\nWe conclude from (4.16) that \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} < a \) . Since \( A \) is an \( a \) -Schwarzian domain, \( f \) is univalent. This is a contradiction, and so \( A \) is linearly locally connected with the constant \( c = 1 + {16}/a \) .
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Yes
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Theorem 5.2. Let \( f \) be meromorphic in a disc. If\n\n\[ \begin{Vmatrix}{S}_{f}\end{Vmatrix} \leq 2 \]\n\nthen \( f \) is univalent. The bound 2 is best possible.
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Proof. Consider functions \( {f}_{n}, n = 1,2,\ldots \), which are meromorphic in the given disc, fix three points of the disc, and have Schwarzians \( \left( {1 - 1/n}\right) {S}_{f} \) ; by Theorem 1.1 such functions exist. Since \( \begin{Vmatrix}{S}_{{f}_{n}}\end{Vmatrix} < 2 \), every \( {f}_{n} \) is univalent owing to Theorem 5.1. They form a normal family, and the limit of a locally uniformly convergent subsequence is a univalent function. Since the limit function shares the same Schwarzian derivative with \( f \), we conclude that \( f \) is univalent.\n\nIn order to prove that the bound 2 cannot be replaced by a larger number, consider the analytic function \( z \rightarrow f\left( z\right) = {z}^{i\varepsilon },\varepsilon > 0 \), in the upper half-plane. Then \( {S}_{f}\left( z\right) = \left( {1 + {\varepsilon }^{2}}\right) {\left( 2{z}^{2}\right) }^{-1} \), and so \( \begin{Vmatrix}{S}_{f}\end{Vmatrix} = 2\left( {1 + {\varepsilon }^{2}}\right) \) . On the other hand, \( f \) is not univalent for any \( \varepsilon > 0 \) : For instance, \( f \) takes the same value at the points \( i \) and \( i\exp \left( {{2\pi }/\varepsilon }\right) \) of the upper half-plane.
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Yes
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Theorem 5.4. Let \( f \) be meromorphic and satisfy\n\n\[ \n\\left| {{S}_{f}\\left( z\\right) }\\right| < \\frac{2}{{\\left( 1 - {\\left| z\\right| }^{2}\\right) }^{2}}\n\]\n\nin the unit disc. Then \( f \) is univalent and has a homeomorphic extension to the plane.
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Proof. By Theorem 5.2, \( f \) is univalent. The image \( f\\left( D\\right) \) is a Jordan domain if and only if \( f \) has a homeomorphic extension to the plane. Hence, if a homeomorphic extension does not exist, then by Theorem \( {5.3}, f\\left( D\\right) \) is the image of the parallel strip \( A \) under a Möbius transformation. If \( h \) again denotes the conformal mapping \( z \\rightarrow \\tanh \\left( {z/2}\\right) \) of \( A \) onto \( D \), then \( g = f \\circ h \) is a Möbius transformation. It follows from (5.4) that\n\n\[ \n{\\left( 1 - {\\left| z\\right| }^{2}\\right) }^{2}\\left| {{S}_{f}\\left( z\\right) }\\right| = 2\n\]\n\nat every point of \( h\\left( \\mathbb{R}\\right) \) . This is in contradiction with the hypothesis.
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Yes
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Theorem 1.1. Every point of the universal Teichmüller space can be represented by a real analytic quasiconformal mapping \( f \in F \) or by a real analytic complex dilatation \( \mu \in B \) .
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Proof. The result follows immediately from Theorem I.5.3. (For a complete proof, see II.5.2.)
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No
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Theorem 1.2. The complex dilatations \( \mu \) and \( v \) are equivalent if and only if the conformal mappings \( {f}_{\mu }\left| {H}^{\prime }\right| \) and \( {f}_{v} \mid {H}^{\prime } \) coincide.
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Proof. Suppose first that \( {f}_{\mu }\left| {{H}^{\prime } = {f}_{v}}\right| {H}^{\prime } \) . The mappings \( {f}_{\mu } \circ {\left( {f}^{\mu }\right) }^{-1} \) and \( {f}_{v} \circ {\left( {f}^{v}\right) }^{-1} \) are both conformal in the upper half-plane \( H \), which they map onto the same quasidisc. Because they fix \( 0,1,\infty \), it follows that they agree in \( H \), and hence also on the real axis \( \mathbb{R} \) . Since \( {f}_{\mu } = {f}_{v} \) on \( \mathbb{R} \), we conclude that \( {f}^{\mu } = {f}^{v} \) on \( \mathbb{R} \), i.e., \( \mu \) and \( v \) are equivalent.\n\nAssume, conversely, that \( {f}^{\mu } = {f}^{v} \) on \( \mathbb{R} \) . We define a mapping \( w \) of the plane by the requirements \( w = {f}_{\mu } \circ {f}_{v}^{-1} \) in \( {f}_{v}\left( {{H}^{\prime } \cup \mathbb{R}}\right) \), and \( w = \) \( {f}_{\mu } \circ {\left( {f}^{\mu }\right) }^{-1} \circ {f}^{v} \circ {f}_{v}^{-1} \) in \( {f}_{v}\left( H\right) \) . From the hypothesis \( {f}^{\mu } = {f}^{v} \) on \( \mathbb{R} \) it follows that \( w \) is a homeomorphism of the plane. In addition, \( w \mid {f}_{v}\left( {H}^{\prime }\right) \) is conformal. But so is also \( w \mid {f}_{v}\left( H\right) \), because \( {f}_{\mu } \circ {\left( {f}^{\mu }\right) }^{-1} \) and \( {f}^{v} \circ {f}_{v}^{-1} \) are conformal. Since \( {f}_{v}\left( \mathbb{R}\right) \) is a quasicircle, we infer from Lemma I.6.1 that \( w \) is a Möbius transformation. Owing to the normalization, \( w \) is the identity mapping, and so \( {f}_{\mu } = {f}_{v} \) in \( {H}^{\prime } \) .
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Yes
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Lemma 1.1. Let \( h \) be a normalized quasisymmetric function. Then the sewing problem has a unique normalized pair of solutions.
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Proof. Given a function \( h \in X \), there is a mapping \( {f}^{\mu } \in F \) such that \( {f}^{\mu } \mid \mathbb{R} = h \) . Then\n\n\[ \n{f}_{1} = \left( {{f}_{\mu } \mid H}\right) \circ {\left( {f}^{\mu }\right) }^{-1},\;{f}_{2} = {f}_{\mu } \mid {H}^{\prime }, \]\n\n\nis a solution of the sewing problem. This can be verified immediately.\n\nSuppose that the pair \( \left( {{g}_{1},{g}_{2}}\right) \) is also a normalized solution. Then \( {g}_{2} \mid \mathbb{R} = \) \( {g}_{1} \circ h = {g}_{1} \circ {f}^{\mu } \mid \mathbb{R} \) . Hence, the mapping \( w \) which agrees with \( {g}_{1} \circ {f}^{\mu } \) in \( H \cup \mathbb{R} \) and with \( {g}_{2} \) in \( {H}^{\prime } \) is a homeomorphism of the plane. Off the real axis it is quasiconformal. By Lemma I.6.1, \( w \) is quasiconformal everywhere. Since \( w \) has the same complex dilatation as \( {f}_{\mu } \) and both mappings fix 0,1 and \( \infty \), it follows from the uniqueness theorem (Theorem I.4.2) that \( w = {f}_{\mu } \) . Comparison of the definitions of \( w,{f}_{1} \) and \( {f}_{2} \) then shows that \( {g}_{1} = {f}_{1},{g}_{2} = {f}_{2} \) .\n\nNote that \( {f}_{1} \) and \( {f}_{2} \) map the half-planes onto quasidiscs. Lemma 1.1 is due to Pfluger [2]; in [LV], p. 92, it was proved without the use of the existence theorem for Beltrami equations.
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Yes
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Theorem 1.3. Two points \( \left\lbrack {f}^{\mu }\right\rbrack ,\left\lbrack {f}^{v}\right\rbrack \in T \) are inverse elements of the group \( T \) if and only if the quasidiscs \( {f}_{\mu }\left( H\right) \) and \( {f}_{v}\left( {H}^{\prime }\right) \) are mirror images with respect to the real axis.
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Proof. Assume first that \( \left\lbrack {f}^{\mu }\right\rbrack \) and \( \left\lbrack {f}^{v}\right\rbrack \) are inverse; we can then take \( {f}^{v} = {\left( {f}^{\mu }\right) }^{-1} \) . Let \( {f}_{{\mu }^{ * }} \) be the quasiconformal mapping of the plane which fixes the points \( 0,1,\infty \) and whose complex dilatation \( {\mu }^{ * } \) vanishes in \( H \) and equals \( \bar{\mu }\left( \bar{z}\right) \) at almost all points \( z \in {H}^{\prime } \) . We write \( {g}_{1} = {f}_{{\mu }^{ * }} \mid H \) and denote by \( {g}_{2} \) the unique conformal mapping of \( {H}^{\prime } \) onto \( {f}_{{\mu }^{ * }}\left( {H}^{\prime }\right) \) which keeps \( 0,1,\infty \) fixed. Then \( {g}_{1} \) and \( {g}_{2} \) are normalized conformal mappings of the upper and lower half-planes, respectively, onto complementary quasidiscs.\n\nIn order to study \( {g}_{1}^{-1} \circ {g}_{2} \) on the real axis \( \mathbb{R} \), we continue \( {f}^{\mu } \) by reflection in \( \mathbb{R} \) and use the same notation \( {f}^{\mu } \) for the extended mapping. Then\n\n\[ \n{f}^{\mu } = {g}_{2}^{-1} \circ {f}_{{\mu }^{ * }}\n\]\n\nin \( {H}^{\prime } \), because both sides are normalized quasiconformal self-mappings of \( {H}^{\prime } \) with the same complex dilatation. Hence, on \( \mathbb{R} \)\n\n\[ \n{g}_{1}^{-1} \circ {g}_{2} = {\left( {f}^{\mu }\right) }^{-1} = {f}^{v}.\n\]\n\nNow set\n\n\[ \n{f}_{1} = {f}_{v} \circ {\left( {f}^{v} \mid H\right) }^{-1},\;{f}_{2} = {f}_{v} \mid {H}^{\prime }.\n\]\n\nThen \( {f}_{1} \) and \( {f}_{2} \) are also normalized conformal mappings of the upper and lower half-planes onto complementary quasidiscs. On the real axis, \( {f}_{1}^{-1} \circ {f}_{2} = {f}^{v} = {g}_{1}^{-1} \circ {g}_{2} \) . We conclude from Lemma 1.1 that \( {g}_{1} = {f}_{1},{g}_{2} = {f}_{2} \) .\n\nFrom the definition of \( {f}_{\mu * } \) it follows that \( {f}_{\mu * }\left( \bar{z}\right) = {\bar{f}}_{\mu }\left( z\right) \) ; one way to verify this is to compute the partial derivatives. Since \( {f}_{v}\left( H\right) = {f}_{1}\left( H\right) = {g}_{1}\left( H\right) = \) \( {f}_{{\mu }^{ * }}\left( H\right) \), we obtain\n\n\[ \n{f}_{v}\left( {H}^{\prime }\right) = {f}_{{\mu }^{ * }}\left( {H}^{\prime }\right) = {\bar{f}}_{\mu }\left( H\right)\n\]\n\nand the first part of the theorem has been proved.\n\nAfter this the converse is easily established. Suppose that \( {\bar{f}}_{\mu }\left( H\right) = {f}_{v}\left( {H}^{\prime }\right) \) . By what was just proved, there is a quasiconformal mapping \( {f}_{\lambda } \), where \( \lambda \) is determined by \( {f}^{\lambda } = {\left( {f}^{\mu }\right) }^{-1} \), such that \( {\bar{f}}_{\mu }\left( H\right) = {f}_{\lambda }\left( {H}^{\prime }\right) \) . Since the mapping (1.3) is injective, we conclude that \( \lambda \) is equivalent to \( v \) . It follows that \( \left\lbrack {f}^{\mu }\right\rbrack \) and \( \left\lbrack {f}^{v}\right\rbrack \) are inverse elements of \( T \) .
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Yes
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Lemma 2.1. The functions \( \tau ,{\tau }_{1} \) and \( {\tau }_{2} \) are the same.
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Proof. Clearly, \( \tau \leq {\tau }_{1} \) . If \( w \in W \), then \( g = w \circ {f}_{0} \in q \), so that \( {\tau }_{1} \leq {\tau }_{2} \) . Finally, if \( f \in p, g \in q \), then \( g \circ {f}^{-1} \in W \), and so \( {\tau }_{2} \leq \tau \) .
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No
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Theorem 2.1. The universal Teichmüller space is pathwise connected.
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Proof. Consider the origin of \( T \), i.e., the point represented by the function of \( B \) which is identically zero, and an arbitrary point \( p \in T \) represented by \( \mu \) . For \( 0 \leq t \leq 1 \), let \( {p}_{t} \) be the point represented by the function \( {t\mu } \) of \( B \) . Then\n\n\[ \beta \left( {{p}_{{t}_{1}},{p}_{{t}_{2}}}\right) \leq {\begin{Vmatrix}\frac{{t}_{1}\mu - {t}_{2}\mu }{1 - {t}_{1}{t}_{2}{\left| \mu \right| }^{2}}\end{Vmatrix}}_{\infty } \leq \frac{\left| {t}_{1} - {t}_{2}\right| }{1 - \parallel \mu {\parallel }_{\infty }^{2}}. \]\n\nWe see that the mapping \( t \rightarrow {p}_{t} \) is continuous, i.e., it is a path in \( T \) joining the origin to \( p \) .
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Yes
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Theorem 2.2. If \( \mu \) is an extremal complex dilatation for the point \( p \in T \), then\n\n\[ \n{\mu }_{t} = \frac{{\left( 1 + \left| \mu \right| \right) }^{t} - {\left( 1 - \left| \mu \right| \right) }^{t}}{{\left( 1 + \left| \mu \right| \right) }^{t} + {\left( 1 - \left| \mu \right| \right) }^{t}}\frac{\mu }{\left| \mu \right| },\;0 \leq t \leq 1, \n\]\n\n(2.5)\n\nis extremal for the point \( {p}_{t} = \left\lbrack {\mu }_{t}\right\rbrack \) . The arc \( t \rightarrow {p}_{t} \) is a geodesic from 0 to \( p \), and\n\n\[ \n\tau \left( {{p}_{t},0}\right) = {t\tau }\left( {p,0}\right) \n\]\n\n(2.6)
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Proof. From (2.5) we see that \( {\mu }_{t}\left( z\right) \) is the point which divides the hyperbolic length (in the unit disc) of the line segment from 0 to \( \mu \left( z\right) \) in the ratio \( t : \left( {1 - t}\right) \) (cf. formula (4.16) in I.4.7).\n\nIf \( {f}^{\mu } \) has maximal dilatation \( K \), then by Theorem I.4.7, the mapping \( {f}^{{\mu }_{t}} \) has maximal dilatation \( {K}^{t} \) and \( {f}^{\mu } \circ {\left( {f}^{{\mu }_{t}}\right) }^{-1} \) has maximal dilatation \( {K}^{1 - t} \) . Suppose that \( w \in {p}_{t} \) . Then \( \varphi = {f}^{\mu } \circ {\left( {f}^{{\mu }_{t}}\right) }^{-1} \circ w \in p \), and so \( K \leq {K}_{\varphi } \leq {K}^{1 - t}{K}_{w} \) . Consequently, \( {K}_{w} \geq {K}^{t} \) . We conclude that \( {\mu }_{t} \) is extremal for the point \( {p}_{t} \) . This reasoning also shows that \( \tau \left( {{p}_{t},0}\right) = \frac{1}{2}\log {K}^{t} = {t\tau }\left( {p,0}\right) \), i.e., the validity of (2.6).\n\nSince \( {f}^{\mu } \circ {\left( {f}^{{\mu }_{t}}\right) }^{-1} \) has maximal dilatation \( {K}^{1 - t} \), we conclude that \( \tau \left( {{p}_{t}, p}\right) \leq \) \( \left( {1 - t}\right) \tau \left( {p,0}\right) \) . Consequently, \( \tau \left( {0,{p}_{t}}\right) + \tau \left( {{p}_{t}, p}\right) = \tau \left( {0, p}\right) \) for every \( t \) . Finally, if we repeat the above argument for an arbitrary subarc of \( t \rightarrow {p}_{t} \), we see that \( t \rightarrow {p}_{t} \) is a geodesic.\n\nSince the extremal \( \mu \) need not be unique (cf. I.5.7), we cannot conclude that the geodesic \( t \rightarrow \left\lbrack {\mu }_{t}\right\rbrack \) is unique.
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Yes
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Theorem 2.3. The universal Teichmüller space is complete.
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Proof. In view of statement \( {2}^{ \circ } \) in Lemma 2.2, it is enough to observe that if a Cauchy sequence contains a convergent subsequence, then the sequence itself is convergent.
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No
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Theorem 3.1. The group isomorphism\n\n\\[ \n\\left\\lbrack f\\right\\rbrack \\rightarrow f \\mid \\mathbb{R} \n\\]\n\n(3.1)\n\nis a homeomorphism of \\( \\left( {T,\\tau }\\right) \\) onto \\( \\left( {X,\\rho }\\right) \\) .
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Proof. We proved in 1.1 that (3.1) is a bijection of \\( T \\) onto \\( X \\) . From (2.4) and the left-hand inequality (5.10) in I.5.7 it follows that\n\n\\[ \n\\rho \\left( {{f}_{1}\\left| {\\mathbb{R},{f}_{2}}\\right| \\mathbb{R}}\\right) \\leq \\tau \\left( {\\left\\lbrack {f}_{1}\\right\\rbrack ,\\left\\lbrack {f}_{2}\\right\\rbrack }\\right) .\n\\]\n\n(3.2)\n\nHence (3.1) is continuous. From Lemma I.5.5 (or from the right-hand inequality (5.10) in I.5.7) we conclude that the inverse of (3.1) is continuous.\n\nFrom the double inequality (5.10) in I.5.7 we can draw another conclusion: The space \\( \\left( {X,\\rho }\\right) \\) is complete. For we conclude from the right-hand inequality (5.10) that the preimage of a Cauchy sequence in \\( \\left( {X,\\rho }\\right) \\) is a Cauchy sequence in \\( \\left( {T,\\tau }\\right) \\) . The inequality (3.2) then shows that (3.1) maps a convergent sequence of \\( \\left( {T,\\tau }\\right) \\) onto a convergent sequence of \\( \\left( {X,\\rho }\\right) \\) .\n\nSuppose that \\( h,{h}_{n} \\in X, n = 1,2,\\ldots \\), and that \\( \\lim \\rho \\left( {{h}_{n}, h}\\right) = 0 \\) . Then \\( {h}_{n} \\rightarrow h \\) locally uniformly in the euclidean metric. For by Lemma I.5.1, \\( \\left\\{ {h}_{n}\\right\\} \\) is a normal family. If \\( \\widetilde{h} \\) is the limit of a convergent subsequence \\( \\left( {h}_{{n}_{i}}\\right) \\) of \\( \\left( {h}_{n}\\right) \\), we have \\( {K}_{h \\circ {h}^{-1}}^{ * } \\leq \\lim {K}_{{h}_{n}, \\circ {h}^{-1}}^{ * } = 1 \\) . Hence \\( \\widetilde{h} = h \\) . Since every convergent subsequence of \\( \\left( {h}_{n}\\right) \\) has the limit \\( h \\), the sequence itself tends to \\( h \\) .
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Yes
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Theorem 3.2. The universal Teichmüller space is contractible.
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Proof. Every point of \( T \) is an equivalence class \( \left\lbrack {s\left( h\right) }\right\rbrack, h \in X \) . We show that\n\n\[ \left( {\left\lbrack {s\left( h\right) }\right\rbrack, t}\right) \rightarrow \left\lbrack {\left( {1 - t}\right) s\left( h\right) }\right\rbrack \]\n\n(3.3)\n\ndeforms \( T \) continuously to the point 0 as \( t \) increases from 0 to 1 .\n\nIn proving this, we make use of Theorem 3.1 which says that \( \left\lbrack \mu \right\rbrack \rightarrow {f}^{\mu } \mid \mathbb{R} \) is a homeomorphism of \( \left( {T,\tau }\right) \) onto \( \left( {X,\rho }\right) \) . It means that instead of (3.3), we can consider the induced mapping\n\n\[ \left( {h, t}\right) \rightarrow {f}^{\left( {1 - t}\right) s\left( h\right) } \mid \mathbb{R} \]\n\n(3.4)\n\nof \( X \times \left\lbrack {0,1}\right\rbrack \) into \( X \) . Clearly, \( \left( {h,0}\right) \rightarrow h \) and \( \left( {h,1}\right) \rightarrow \) identity. The theorem follows if we prove that (3.4) is continuous.\n\nThe mapping (3.4) is the composition of the three mappings\n\n\[ \left( {h, t}\right) \rightarrow \left( {s\left( h\right), t}\right) ,\;\left( {s\left( h\right), t}\right) \rightarrow \left( {1 - t}\right) s\left( h\right) ,\;\left( {1 - t}\right) s\left( h\right) \rightarrow {f}^{\left( {1 - t}\right) s\left( h\right) } \mid \mathbb{R}. \]\n\nThe first one is continuous, because we just proved that \( h \rightarrow s\left( h\right) \) is a continuous map of \( X \) into \( B \) . The second maps \( B \times \left\lbrack {0,1}\right\rbrack \) continuously into \( B \) , since \( {\begin{Vmatrix}\left( 1 - {t}_{1}\right) s\left( {h}_{1}\right) - \left( 1 - {t}_{2}\right) s\left( {h}_{2}\right) \end{Vmatrix}}_{\infty } \leq {\begin{Vmatrix}s\left( {h}_{1}\right) - s\left( {h}_{2}\right) \end{Vmatrix}}_{\infty } + \left| {{t}_{1} - {t}_{2}}\right| \) . Finally, the third mapping is continuous, because we showed that \( \mu \rightarrow {f}^{\mu } \mid \mathbb{R} \) maps \( B \) continuously into \( X \) . Hence,(3.4) is a continuous contraction of \( X \) to a point, and (3.3) has the same property with respect to \( T \) .
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Yes
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Theorem 3.3. The universal Teichmüller space is not a topological group.
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Proof. The theorem follows if we find an \( \left\lbrack f\right\rbrack \in T \) and a sequence of points \( \left\lbrack {g}_{n}\right\rbrack \in T \), such that \( \left\lbrack {g}_{n}\right\rbrack \) tends to \( \left\lbrack g\right\rbrack \) but \( \left\lbrack {f \circ {g}_{n}}\right\rbrack \) does not tend to \( \left\lbrack {f \circ g}\right\rbrack \) . Because the mapping (3.1) is a group isomorphism and a homeomorphism, the counterexample can be constructed in \( X \) . We follow a suggestion of P. Tukia.\n\nIn order to simplify notation we write \( f \) instead of \( f \mid \mathbb{R} \) . We define \( f \) as follows: \( f\left( x\right) = x \) if \( x \geq 0, f\left( x\right) = x/2 \) if \( - 2 \leq x < 0 \), and \( f\left( x\right) = x + 1 \) if \( x < - 2 \) . Then \( f \) is a 2-quasisymmetric function of \( X \) . Set \( {l}_{n}\left( x\right) = x \) if \( x \geq 0 \) and \( {l}_{n}\left( x\right) = \left( {1 + 1/n}\right) x \) if \( x < 0, n = 1,2,\ldots \) Then \( {l}_{n} \in X \) is \( \left( {1 + 1/n}\right) \) -quasisymmetric, and therefore \( {\left( 1 + 1/n\right) }^{2} \) -quasiconformal (cf. I.5.3). If \( \iota \) denotes the identity mapping of \( \mathbb{R} \) onto itself, we thus have\n\n\[ \rho \left( {{\iota }_{n},\iota }\right) \leq \log \left( {1 + 1/n}\right) \]\n\nLet us define \( {g}_{n} = {\iota }_{n} \circ {f}^{-1} \) . Because \( \rho \left( {{g}_{n},{f}^{-1}}\right) = \rho \left( {{\iota }_{n},\iota }\right) \), we deduce that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\rho \left( {{g}_{n},{f}^{-1}}\right) = 0 \]\n\n(3.5)\n\nWe prove that \( f \circ {g}_{n} = f \circ {\iota }_{n} \circ {f}^{-1} \) (which converges to the identity \( \iota \) pointwise) does not tend to \( f \circ \left( {\lim {g}_{n}}\right) = \iota \) in the \( \rho \) -metric.\n\nDirect calculation yields\n\n\[ f\left( {{g}_{n}\left( x\right) }\right) = \left\{ \begin{array}{ll} \left( {1 + 1/n}\right) x & \text{ if } - n/\left( {n + 1}\right) \leq x < 0, \\ 2\left( {1 + 1/n}\right) x + 1 & \text{ if } - 1 \leq x < - n/\left( {n + 1}\right) . \end{array}\right. \]\n\nIt follows that \( f \circ {g}_{n} \) has a quasisymmetry constant \( \geq 2 \) for every \( n \) . Consequently,\n\n\[ \rho \left( {f \circ {g}_{n},\iota }\right) \geq \frac{1}{2}\log {\lambda }^{-1}\left( 2\right) \]\n\nIn conjunction with (3.5), this shows that \( \left( {X,\rho }\right) \), and hence \( \left( {T,\tau }\right) \), is not a topological group.\n\nWe see that, unlike the right translation, the left translation \( \left\lbrack f\right\rbrack \rightarrow \left\lbrack {{f}_{0} \circ f}\right\rbrack \) , \( {f}_{0} \) fixed, need not be continuous in \( T \) .
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Yes
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Theorem 4.1. The mapping\n\n\\[ \n\\left\\lbrack \\mu \\right\\rbrack \\rightarrow {S}_{{f}_{\\mu } \\mid H} \n\\]\n\n(4.7)\n\nis a homeomorphism of the universal Teichmüller space onto its image in \\( Q \\) .
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Proof. We noted already in 4.1 that (4.7) is well defined in \\( T \\) . If \\( \\left\\lbrack \\mu \\right\\rbrack \\) and \\( \\left\\lbrack v\\right\\rbrack \\) have the same image, it follows from the normalization that \\( {f}_{\\mu }\\left| {H = {f}_{v}}\\right| H \\), i.e., \\( \\mu \\) and \\( v \\) are equivalent. Hence (4.7) is injective. Inequality (4.4) shows that (4.7) is continuous, and (4.6) that its inverse is continuous.
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Yes
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Theorem 4.2. The set \( T\left( 1\right) \) is the interior of \( U \) .
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Proof. We prove first that \( T\left( 1\right) \) is an open subset of \( Q \) . Fix an arbitrary point \( {S}_{f} \) of \( T\left( 1\right) \) . For \( {S}_{h} \in Q \) we write \( g = h \circ {f}^{-1} \), and conclude that \( g \) is meromorphic in the quasidisc \( f\left( H\right) \) . By Theorem II.4.1, there exists a positive constant \( \varepsilon \) such that if \( {\begin{Vmatrix}{S}_{g}\end{Vmatrix}}_{f\left( H\right) } < \varepsilon \), then \( g \) is univalent in \( f\left( H\right) \) and has a quasicon-formal extension to the plane. Now choose \( {S}_{h} \in Q \) such that \( {\begin{Vmatrix}{S}_{h} - {S}_{f}\end{Vmatrix}}_{H} < \varepsilon \) . Then\n\n\[ \n{\begin{Vmatrix}{S}_{g}\end{Vmatrix}}_{f\left( H\right) } = {\begin{Vmatrix}{S}_{h} - {S}_{f}\end{Vmatrix}}_{H} < \varepsilon .\n\]\n\nBecause \( h = g \circ f \), we conclude that \( {S}_{h} \in T\left( 1\right) \) . It follows that \( T\left( 1\right) \) is open.\n\nSince \( T\left( 1\right) \subset U \), the proof will be complete if we show that int \( U \subset T\left( 1\right) \) . Choose a point \( {S}_{f} \in \) int \( U \) . We then have an \( \varepsilon > 0 \) such that\n\n\[ \nV = \left\{ {\varphi \in Q \mid \begin{Vmatrix}{\varphi - {S}_{f}}\end{Vmatrix} \leq \varepsilon }\right\} \subset U.\n\]\n\nLet \( g \) be an arbitrary meromorphic function in the domain \( f\left( H\right) \), with the property \( {\begin{Vmatrix}{S}_{g}\end{Vmatrix}}_{f\left( H\right) } \leq \varepsilon \) . If \( h = g \circ f \), then\n\n\[ \n{\begin{Vmatrix}{S}_{h} - {S}_{f}\end{Vmatrix}}_{H} = {\begin{Vmatrix}{S}_{g}\end{Vmatrix}}_{f\left( H\right) } \leq \varepsilon .\n\]\n\nIt follows that \( {S}_{h} \in V \subset U \), i.e., \( h \) is univalent in \( H \) . But then \( g = h \circ {f}^{-1} \) is univalent in \( f\left( H\right) \) . What we have proved is that \( f\left( H\right) \) is an \( \varepsilon \) -Schwarzian domain. Hence, by Theorem II.4.2, the domain \( f\left( H\right) \) is a quasidisc. We conclude that \( {S}_{f} \in T\left( 1\right) \) (cf. Lemma I.6.2, statement \( {3}^{ \circ } \) ) as we wished to show.
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Yes
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Theorem 4.3. The closure of \( T\left( 1\right) \) is a proper subset of \( U \) .
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Proof. Let \( G \) be the domain defined above and \( \varepsilon > 0 \) the associated constant. If \( h \) is a conformal mapping of the upper half-plane onto \( G \), we prove that \( {S}_{h} \) does not lie in the closure of \( T\left( 1\right) \) .\n\nConsider an arbitrary point \( {S}_{w} \) of the neighborhood \( \left\{ {\varphi \in Q \mid {\begin{Vmatrix}\varphi - {S}_{h}\end{Vmatrix}}_{H} < \varepsilon }\right\} \) . For \( f = w \circ {h}^{-1} \) we then have \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{G} = {\begin{Vmatrix}{S}_{w} - {S}_{h}\end{Vmatrix}}_{H} < \varepsilon \) . Therefore, either \( f \) is not univalent or \( f \) is univalent but \( f\left( G\right) = w\left( H\right) \) is not a Jordan domain. It follows that \( {S}_{w} \) is not in \( T\left( 1\right) \) .
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Yes
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Lemma 5.2. Let \( A \) be a quasidisc which is contained in a domain \( {B}_{k} \) Möbius equivalent to the sector \( {A}_{k} = \{ z \mid \left| {\arg z}\right| < {k\pi }/2\} \) . If \( 0 < k \leq 1 \), assume that a vertex \( v \) of \( {B}_{k} \) lies on \( \partial A \) . Then\n\n\[ \n{\sigma }_{I}\left( A\right) \leq 2{k}^{2} \n\]\n\nIf \( 1 < k < 2 \), assume that there are points \( {z}_{1} \) and \( {z}_{2} \) in \( \partial A \) such that for a Möbius transformation \( g \) mapping \( {B}_{k} \) onto \( {A}_{k}, g\left( {z}_{1}\right) = {e}^{{ik\pi }/2}, g\left( {z}_{2}\right) = {e}^{-{ik\pi }/2} \) . Then\n\n\[ \n{\sigma }_{I}\left( A\right) \leq {4k} - 2{k}^{2} \n\]
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Proof. Suppose first that \( 0 < k \leq 1 \) . Let \( g \) be a Möbius transformation mapping \( {B}_{k} \) onto \( {A}_{k} \) with \( g\left( v\right) = 0 \) . Set \( f\left( z\right) = \log g\left( z\right) \) . Then \( f\left( A\right) \) is not a quasidisc, and so by (5.5), \( {\sigma }_{I}\left( A\right) \leq {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} \) . By the monotonicity of the hyperbolic metric (formula (1.2) in I.1.1), \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} \leq {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{{B}_{k}} = 2{k}^{2} \) .\n\nSuppose next that \( 1 < k < 2 \) . From the proof of Theorem 5.2 we deduce the existence of a conformal mapping \( f \) of \( {A}_{k} \) such that \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{{A}_{k}} = {4k} - {k}^{2} \) and that \( f\left( {e}^{{ik\pi }/2}\right) = f\left( {e}^{-{ik\pi }/2}\right) = \infty \) . Then \( f\left( {g\left( A\right) }\right) \) is not a Jordan domain, and by reasoning as in the case \( 0 < k \leq 1 \), we arrive at the desired estimate.
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Yes
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Lemma 5.3. Let \( A \) be a quasidisc. If every two-point subset of \( A \) is contained in the closure of a quasidisc \( B \subset A \) for which \( {\sigma }_{I}\left( B\right) \geq m \), then\n\n\[ \n{\sigma }_{I}\left( A\right) \geq m\text{.} \n\]
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Proof. Let an \( \varepsilon > 0 \) be given. There exists a meromorphic function \( f \) in \( A \) for which \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} < {\sigma }_{I}\left( A\right) + \varepsilon \) but which is not univalent. Let \( {z}_{1} \) and \( {z}_{2} \) be two different points of \( A \) such that \( f\left( {z}_{1}\right) = f\left( {z}_{2}\right) \), and \( B \subset A \) a quasidisc such that \( \left\{ {{z}_{1},{z}_{2}}\right\} \subset \bar{B} \) and \( {\sigma }_{I}\left( B\right) \geq m \) . Since either \( f \) is not univalent in \( B \) or else \( f\left( B\right) \) is not a quasidisc, \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{B} \geq {\sigma }_{I}\left( B\right) \) . By the monotonicity of the hyperbolic metric, \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} \geq {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{B} \) . Hence \( {\sigma }_{I}\left( A\right) > m - \varepsilon \), and the lemma follows.
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Yes
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Theorem 5.3. For all domains A conformally equivalent to a disc,\n\n\[ \n{\sigma }_{I}\left( A\right) \leq 2\text{.} \n\]\n\nEquality holds if and only if \( A \) is a disc.
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Proof. Let \( A \) be an arbitrary quasidisc. Every Jordan domain is Möbius equivalent to a subdomain of \( H \) having 0 and \( \infty \) as boundary points. We may assume, therefore, that \( A \) itself is such a domain.\n\nIn \( A \), we consider the function \( z \rightarrow f\left( z\right) = \log z \), for which \( {S}_{f}\left( z\right) = 1/\left( {2{z}^{2}}\right) \) . From the monotonicity of the hyperbolic metric it follows that \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} \leq 2 \) . Because \( f \) maps both 0 and \( \infty \) to infinity, \( f\left( A\right) \) is not a Jordan domain.\n\nHence, we obtain (5.13) from the characterization (5.5) of the inner radius.\n\nThe same idea, in a refined form, can be used to prove that \( {\sigma }_{I}\left( A\right) = 2 \) only if \( A \) is a disc (Lehtinen [2]). We now assume that \( A \subset H \) has two finite boundary points on the real axis. If \( A \) is not \( H \), there are two finite points in \( \partial A \cap \mathbb{R} \) such that the open interval on \( \mathbb{R} \) between these points lies in the complement of \( \partial A \) . A simple geometric argument shows that \( A \) then lies in a non-convex sector both of whose sides contain a point of \( \partial A \) at an equal distance from the vertex (for the details, see Lehtinen [2]). Therefore, we may assume that \( A \) lies in an angle \( {A}_{k} = \{ z \mid 0 < \arg z < {k\pi }\} ,1 < k < 2 \), such that the points 1 and \( {e}^{k\pi i} \) are on the boundary of \( A \) .\n\nInstead of the logarithm, we now consider the extremal mapping \( f \) of the sectoral domain \( {A}_{k} \) exhibited in the proof of Theorem 5.2. Since \( f\left( 1\right) = f\left( {e}^{k\pi i}\right) \) , the image of \( A \) under \( f \mid A \) is not a Jordan domain. It follows from (5.5), the monotonicity of the hyperbolic metric, and Theorem 5.2, that\n\n\[ \n{\sigma }_{I}\left( A\right) \leq {\begin{Vmatrix}{S}_{f \mid A}\end{Vmatrix}}_{A} \leq {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{{A}_{k}} = {4k} - 2{k}^{2} < 2. \n\]
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Yes
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Theorem 1.1. Every orientable \( {C}^{2} \) -surface in \( {\mathbb{R}}^{3} \) can be made into a Riemann surface.
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Proof. Let \( S \) be an orientable \( {C}^{2} \) -surface. Consider an arbitrary local parameter of \( S \) inducing local coordinates \( z \) in a domain \( A \) of the complex plane. The theorem follows if we can transform the \( z \) -coordinates diffeomorphically so that the new coordinates are isothermal.\n\nExpressed in terms of \( z \), the line element of \( S \) is of the form (1.4). Here \( \mu \) is continuously differentiable and by (1.5), we have \( \sup \left| {\mu \left( z\right) }\right| < 1 \) in every relatively compact subdomain of \( A \) . Let \( z \rightarrow w \) be a quasiconformal mapping of such a subdomain with complex dilatation \( \mu \) . By the Existence theorem I.4.4 such a mapping \( w \) exists, and by the remark in I.4.5, \( w \) is continuously differentiable and \( \partial w\left( z\right) \neq 0 \) everywhere. Comparison of\n\n\[ \left| {dw}\right| = \left| {\partial {wdz} + \bar{\partial }{wd}\bar{z}}\right| = \left| {\partial w}\right| \left| {{dz} + {\mu d}\bar{z}}\right| \]\n\nwith (1.4) shows that\n\n\[ {ds} = \frac{\lambda }{\left| \partial w\right| }\left| {dw}\right| \]\n\nWe see that the \( w \) -coordinates are isothermal, and the theorem is proved.
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No
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Theorem 2.1 (Monodromy Theorem). Let \( \left( {W, f}\right) \) be an unlimited covering surface of a surface \( S \), and \( {\gamma }_{0} \) and \( {\gamma }_{1} \) homotopic paths on \( S \) . Then the lifts of \( {\gamma }_{0} \) and \( {\gamma }_{1} \) on \( W \) from the same initial point have the same terminal point and they are homotopic.
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Suppose, in particular, that the surface \( S \) is simply connected, i.e., that the fundamental group of \( S \) is trivial. In this case the monodromy theorem yields an interesting corollary:\n\nIf \( \left( {W, f}\right) \) is an unlimited covering surface of a simply connected surface \( S \) , then the mapping \( f : W \rightarrow S \) is a homeomorphism.\n\nFor since the projection \( f \) is continuous, open and surjective, it is enough to show that \( f \) is injective. Assume that there are two points \( a \) and \( b \) of \( W \) such that \( f\left( a\right) = f\left( b\right) \) . A path \( \gamma \) from \( a \) to \( b \) then has a projection on \( S \) which is a closed curve. This is homotopic to zero, since \( S \) is simply connected. By the monodromy theorem, \( \gamma \) terminates at the same point as the constant path \( t \rightarrow a \) . Hence \( a = b \) .
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Yes
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Lemma 2.1. Let \( \left( {W, f}\right) \) be a covering surface of \( S \) . For every \( p \in W \), there are parameter discs \( U \ni p \) and \( f\left( U\right) \), with local parameters \( k \) and \( h \) normalized by \( k\left( p\right) = h\left( {f\left( p\right) }\right) = 0 \), such that in \( U \), \[ h \circ f = {k}^{n} \] where \( n \) is a natural number.
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The proof is given in Ahlfors-Sario [1], p. 40. Conversely, if \( f : W \rightarrow S \) is a continuous mapping and the above condition holds, we conclude immediately that \( \left( {W, f}\right) \) is a covering surface of \( S \) . Thus this condition characterizes covering surfaces.
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No
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Theorem 2.2. If the projection mapping \( f : W \rightarrow S \) is surjective and the covering group \( G \) of \( W \) over \( S \) is transitive, then \( W/G \) and \( S \) are homeomorphic.
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Proof. We write \( \left\lbrack p\right\rbrack \in W/G \) for the equivalence class containing the point \( p \in W \) and prove that\n\n\[ \left\lbrack p\right\rbrack \rightarrow f\left( p\right) \]\n\n(2.2)\n\nis a homeomorphism of \( W/G \) onto \( S \) . First, it follows from \( f = f \circ g, g \in G \) , that (2.2) is well defined in \( W/G \) . It is surjective, because \( f : W \rightarrow S \) is onto, and injective, because \( G \) is transitive. Its continuity follows from the continuity of \( f : W \rightarrow S \), and the continuity of its inverse from the fact that \( f : W \rightarrow S \) is locally homeomorphic.
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Yes
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Theorem 2.3. The covering group of a universal covering surface \( W \) over a surface \( S \) is transitive.
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Proof. Suppose that \( a \) and \( {a}^{\prime } \) are points of \( W \) which lie over the same point of \( S \) . Choose a point \( p \in W \), join \( a \) to \( p \) by a path on \( W \), project this path onto \( S \), and lift the projection back, but from the point \( {a}^{\prime } \) . Let \( {p}^{\prime } \) be the terminal point of this lift. We define \( g \) by the condition \( g\left( p\right) = {p}^{\prime } \), and check that \( g \) is well defined and a cover transformation of \( W \) over \( S \) . Hence \( a \) and \( {a}^{\prime } \) are equivalent under the covering group.
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Yes
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Theorem 2.4. The covering group of a universal covering surface of \( S \) is isomorphic to the fundamental group of \( S \) .
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Proof. Given a point \( a \in W \), let \( \gamma \) be a closed path on \( S \) from \( f\left( a\right) \), and \( b \in W \) the terminal point of the lift of \( \gamma \) from \( a \) . Then \( a \) and \( b \) both lie over \( f\left( a\right) \) . By Theorem 2.3, there is a unique cover transformation \( {g}_{\gamma } \) with the property \( {g}_{\gamma }\left( a\right) = b \) . It is easy to verify that \( \left\lbrack \gamma \right\rbrack \rightarrow {g}_{\gamma } \) is the desired group isomorphism (cf. Ahlfors-Sario [1], p. 38).
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Yes
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Theorem 2.6. Let \( W \) be a surface, \( G \) a properly discontinuous fixed point free group of homeomorphisms of \( W \) onto itself, and \( f : W \rightarrow W/G \) the canonical projection. Then\n\n1. \( W/G \) is a surface,\n\n2. \( \left( {W, f}\right) \) is an unlimited covering surface of \( W/G \) ,\n\n3. \( G \) is the (transitive) covering group of Wover \( W/G \) .
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Proof. By definition, \( f \) is continuous. If \( A \subset W \), then \( {f}^{-1}\left( {f\left( A\right) }\right) = \cup g\left( A\right) \) , \( g \in G \), from which we conclude that \( f \) is open.\n\nIn order to prove that \( W/G \) is a Hausdorff space we consider two different points \( f\left( a\right) \) and \( f\left( b\right) \) of \( W/G \) . Since \( G \) is properly discontinuous, there exists a compact neighborhood \( B \) of \( b \) which does not contain any point \( g\left( a\right), g \in G \) . After this we conclude the existence of a compact neighborhood \( A \) of \( a \) such that \( A \cap g\left( B\right) \) is empty for every \( g \in G \) . Then \( {g}_{1}\left( A\right) \cap {g}_{2}\left( B\right) = \varnothing \) for all \( {g}_{1},{g}_{2} \in G \), and it follows that \( f\left( A\right) \) and \( f\left( B\right) \) are disjoint neighborhoods of \( f\left( a\right) \) and \( f\left( b\right) \) .\n\nClearly \( W/G \) is connected and has a countable base for topology. In order to find local parameters we fix a point \( p \in W \) . Since \( G \) is properly discontinuous and fixed point free, there exists an open neighborhood \( U \) of \( p \) such that \( g\left( U\right) \cap U = \varnothing \) for all mappings \( g \in G \) different from the identity. Then \( f \mid U \) is injective, and if \( U \) is so small that it lies in the domain of a local parameter \( h \) of \( W \), then \( h \circ {\left( f \mid U\right) }^{-1} \) maps the open set \( f\left( U\right) \) in \( W/G \) homeo-morphically onto an open set in the plane. Since \( f : W \rightarrow W/G \) is surjective, it follows that \( W/G \) is a surface. Also, \( \left( {W, f}\right) \) is a smooth covering surface of \( W/G \) .\n\nFrom the definition it is clear that every \( g \in G \) is a cover transformation. Conversely, let \( w \) be a cover transformation and \( p \in W \) . Then there is a \( g \in G \) such that \( g\left( p\right) = w\left( p\right) \), for otherwise we would have \( f\left( {w\left( p\right) }\right) \neq f\left( p\right) \) . Hence \( w = g \).\n\nSince \( G \) is a transitive covering group, it is not difficult to show that \( \left( {W, f}\right) \) is an unlimited covering surface of \( W/G \) (cf. Ahlfors-Sario [1], p. 29).
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Yes
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Theorem 3.1. Let \( S \) be a Riemann surface and \( \left( {W, f}\right) \) a smooth covering surface of \( S \) . Then \( W \) carries a unique conformal structure which makes the projection mapping fanalytic.
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Proof. Let \( H \) be the conformal structure of \( S \) . For every point \( p \in W \) we choose a neighborhood \( U \) of \( p \) such that \( f \mid U \) is injective and \( f\left( U\right) \) is contained in the domain of some \( h \in H \) . Then the atlas \( \{ h \circ \left( {f \mid U}\right) \mid p \in W\} \) defines a conformal structure for \( W \), and \( f \) is analytic with respect to this structure. We say that this conformal structure of \( W \) is obtained by lifting the conformal structure of \( S \) . If the projection \( f : W \rightarrow S \) is analytic with respect to a conformal structure of \( W \), then the condition which expresses this fact shows directly that this structure is the same as the lifted structure. Thus the uniqueness assertion in the theorem follows.
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Yes
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Theorem 3.2. Let \( W \) be a Riemann surface, \( G \) a properly discontinuous fixed point free group of conformal self-mappings of \( W \), and \( f : W \rightarrow W/G \) the canonical projection. Then the surface \( W/G \) carries a unique conformal structure which lifts to the original conformal structure of \( W \) .
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This follows immediately from the way the local parameters of \( W/G \) were defined in the proof of Theorem 2.6. In the situation of Theorem 3.2, the conformal structure of \( W \) is said to have been projected to \( W/G \) . If \( W \) is a given Riemann surface, we always regard the quotient \( W/G \) as the Riemann surface with the projected structure.
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Yes
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Theorem 3.4. Given an arbitrary Riemann surface \( S \), let \( D \) be its universal covering surface, and \( G \) the covering group of \( D \) over \( S \). Then \( S \) is conformally equivalent to the Riemann surface \( D/G \).
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Proof. It follows from Theorems 2.3,2.5, and 3.2 that the quotient \( D/G \) is a Riemann surface with the projected conformal structure. By Theorem 2.2, the mapping (2.2) is a homeomorphism of \( D/G \) onto \( S \). It is conformal, because the conformal structure of \( S \) is also obtained by projection from \( D \).
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Yes
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Theorem 3.5. Two homeomorphisms \( {\varphi }_{i} : {S}_{1} \rightarrow {S}_{2}, i = 0,1 \), induce the same group isomorphisms if and only if they are homotopic.
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Proof. Assume first that \( {\varphi }_{0} \) is homotopic to \( {\varphi }_{1} \) . Let \( h \) be a homotopy from \( {\varphi }_{0} \) to \( {\varphi }_{1} \) and \( {f}_{t} \) a lift of \( h\left( {., t}\right) ,0 \leq t \leq 1 \), such that \( {f}_{t} \) is a homotopy between \( {f}_{0} \) and \( {f}_{1} \) .\n\nChoose \( g \in {G}_{1} \) and \( z \in D \), and consider the two paths \( t \rightarrow {f}_{t}\left( {g\left( z\right) }\right) \) and \( t \rightarrow \left( {{f}_{0} \circ g \circ {f}_{0}^{-1}}\right) \left( {{f}_{t}\left( z\right) }\right) \) . Both have the same initial point \( {f}_{0}\left( {g\left( z\right) }\right) \) and the same projection \( t \rightarrow {\pi }_{2}\left( {{f}_{t}\left( z\right) }\right) \) on \( {S}_{2} \) . Hence they agree, and for \( t = 1 \) we obtain the desired result\n\n\[{f}_{0} \circ g \circ {f}_{0}^{-1} = {f}_{1} \circ g \circ {f}_{1}^{-1}.\n\]\n\nAssume, conversely, that \( {\varphi }_{0} \) and \( {\varphi }_{1} \) have lifts \( {f}_{0} \) and \( {f}_{1} \) such that (3.3) holds for every \( g \in {G}_{1} \) . If \( D \) is the unit disc, we define \( {f}_{t}\left( z\right) ,0 < t < 1 \), as follows: \( {f}_{t}\left( z\right) \) is the point of the hyperbolic geodesic arc joining \( {f}_{0}\left( z\right) \) and \( {f}_{1}\left( z\right) \) which divides the hyperbolic length of this arc in the ratio \( t : \left( {1 - t}\right) \) . Then \( {f}_{t} \) is a homotopy between \( {f}_{0} \) and \( {f}_{1} \) .\n\nUnder the mapping \( \theta \left( g\right) = {f}_{0} \circ g \circ {f}_{0}^{-1}\left( { = {f}_{1} \circ g \circ {f}_{1}^{-1}}\right) \) the endpoints of the arc map to \( {f}_{0}\left( {g\left( z\right) }\right) \) and \( {f}_{1}\left( {g\left( z\right) }\right) \) . But since \( \theta \left( g\right) \) leaves hyperbolic distances invariant, \( \theta \left( g\right) \) maps the point \( {f}_{t}\left( z\right) \) to \( {f}_{t}\left( {g\left( z\right) }\right) \) . Hence, \( \theta \left( g\right) \circ {f}_{t} = {f}_{t} \circ g \) . In other words, all mappings \( {f}_{t},0 \leq t \leq 1 \), determine the same group homomorphism. It follows that \( {\pi }_{2} \circ {f}_{t} \circ {\pi }_{1}^{-1} \) is a well defined mapping, and it is a homotopy between \( {\varphi }_{0} \) and \( {\varphi }_{1} \) .\n\nIf \( D \) is the finite plane, all cover transformations are translations \( z \rightarrow z + b \) . Therefore, the above reasoning remains valid if the hyperbolic metric is replaced by the euclidean.
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Yes
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Lemma 4.1. For a Kleinian group \( G \), every point \( \zeta \in L \) is the cluster point of each orbit \( G\left( z\right) \), with the possible exception of \( z = \zeta \) and one other point \( z \in L \) .
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We first deduce from this lemma that if \( G \) is not elementary, every point of \( L \) is the cluster point of other limit points. Hence, \( L \) is then always a perfect set. It follows that for Möbius groups there is a striking dichotomy: Either the limit set contains at most two points or else it contains uncountably many points.\n\nA second conclusion from Lemma 4.1 is that the limit set of a Kleinian group agrees with the boundary of the set of discontinuity. For we have trivially \( \partial \Omega = \bar{\Omega } \cap L \), so that \( \partial \Omega \subset L \) . On the other hand, we infer from Lemma 4.1 that \( L \subset \bar{\Omega } \) . Hence, \( L \subset \bar{\Omega } \cap L = \partial \Omega \), and we obtain the desired result\n\n\[ L = \partial \Omega \]
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No
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Lemma 4.2. Let \( G \) be a Kleinian group such that \( \Omega \) has an invariant component \( A \) which is a Jordan domain different from a disc. Then \( \partial A \) does not have a tangent at a fixed point of a loxodromic element of \( G \) .
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Proof. Assume that the tangent exists at a fixed point of a loxodromic element \( g \in G \) . We may suppose without loss of generality that the fixed point of \( g \) lies at \( z = 0 \), that the tangent at \( z = 0 \) is the real axis and that \( \infty \) is the repulsive fixed point of \( g \) . Then \( g\left( z\right) = r{e}^{i\theta }z \), where \( 0 < r < 1 \) and \( 0 < \theta < {2\pi } \) . Suppose first that \( \theta \neq \pi \), and set\n\n\[ a = \min \left( {\theta /2,\left| {\pi - \theta }\right| /2,\left( {{2\pi } - \theta }\right) /2}\right) \]\n\nthen \( 0 < a \leq \pi /4 \) . Consider the two angles \( {V}_{a} = \left\{ {\rho {e}^{i\varphi } \mid \varphi \in \left( {-a, a}\right) }\right. \) or \( \varphi \in (\pi - a \) , \( \pi + a),\rho \geq 0\} \) . Since the real axis is a tangent, we have for every \( a > 0 \) a disc \( {D}_{a} \) centered at the origin, such that\n\n\[ \partial A \cap {D}_{a} \subset {V}_{a} \cap {D}_{a} \]\n\n(4.5)\n\nNow choose \( z \in \partial A \cap {D}_{a} \cap {V}_{a}, z \neq 0 \) . Then \( g\left( z\right) \in \partial A \cap {D}_{a} \) . On the other hand, it follows from the definition of \( a \) that \( g\left( z\right) \notin {V}_{a} \) . This contradicts (4.5).\n\nIf \( z \rightarrow g\left( z\right) = - {rz} \) belongs to \( G \), then \( g \circ g \) is a hyperbolic transformation with the same fixed points as \( g \) . A modification of the above proof shows that \( \partial A \) does not have a tangent at a fixed point of a hyperbolic element of \( G \) . This proves the lemma.
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Yes
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Theorem 4.2. The boundary of an invariant component of a quasi-Fuchsian group is either a circle or a Jordan curve which fails to have a tangent on an everywhere dense set.
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Proof. First, if \( A \) denotes an invariant component, we clearly have \( \partial A \subset \partial \Omega \) . From (4.3) we then conclude that \( \partial A \subset L \) . If the group is not Fuchsian, it always contains loxodromic elements (Lehner [1], p. 107). By (4.4), we have in this case \( \partial A \subset {\bar{F}}_{l} \) . Hence, the theorem follows from Lemma 4.2.
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No
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Theorem 5.1. Let \( S \) be a Riemann surface and \( G \) the covering group of the upper half-plane \( H \) over \( S \) . Then \( S \) is compact if and only if the Dirichlet regions of \( G \) are bounded in the hyperbolic metric of \( H \) .
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Proof. Suppose first that \( S \) is compact. Let \( N \) be a Dirichlet region with center \( a \) . We consider the hyperbolic discs \( {D}_{n} = \{ z \mid h\left( {z, a}\right) < n\}, n = 1,2,\ldots \) Their projections on \( S = H/G \) form an open covering of \( S \) . Since \( S \) is compact, there is an \( n \) such that the projection of \( {D}_{n} \) alone covers \( S \) . In other words, for every \( z \in H \) there exists a mapping \( g \in G \) for which \( h\left( {g\left( z\right), a}\right) < n \) . Now if \( z \in N \) , then \( h\left( {z, a}\right) \leq h\left( {g\left( z\right), a}\right) \) for every \( g \in G \) . It follows that \( N \subset {D}_{n} \) . Assume, conversely, that the closure of a Dirichlet region of \( G \) lies in \( H \) . Then \( S \) is the image of a compact set under a continuous mapping and hence compact.
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Yes
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Theorem 5.2. The covering group of the upper half-plane over a compact Riemann surface is finitely generated and of the first kind.
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Proof. Let \( S \) be a compact Riemann surface and \( G \) the covering group of \( H \) over \( S \) . The vertices of a Dirichlet region of \( G \) cannot have a limit point in \( H \) . Hence, by Theorem 5.1, a Dirichlet region for \( G \) has a finite number of sides. We conclude using Theorem 4.1 that \( G \) is finitely generated.\n\nIn order to determine the limit set \( L \) of \( G \), we consider an arbitrary point \( x \) of the real axis and set \( U = \{ \zeta \in H\left| \right| \zeta - x \mid < r\} \) . The hyperbolic distance from the point \( x + {iy} \in U \) to the semicircle \( \left| {\zeta - x}\right| = r \) tends to \( \infty \) as \( y \rightarrow 0 \) . On the other hand, by Theorem 5.1 the Dirichlet region containing \( x + {iy} \) has a uniformly bounded hyperbolic diameter for every \( y > 0 \) . It follows that \( U \) contains a Dirichlet region for every \( r > 0 \) . Consequently, \( x \in L \), and so \( L \) is the whole real axis.
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Yes
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Theorem 5.4 (Riemann-Roch Theorem). On a compact Riemann surface of genus \( p \), every divisor \( D \) satisfies the equation\n\n\[ \n\dim D = \dim \left( {-D - {D}_{1}}\right) - \deg D - p + 1.\n\]
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Let us first apply (5.6) for \( D = - {D}_{1} \) . Then, by (5.5) and (5.2), \( p = 1 + \) \( \deg {D}_{1} - p + 1 \), so that \( \deg {D}_{1} = {2p} - 2 \) . By our previous remark, we have\n\n\[ \n\deg {D}_{{\varphi }_{1}} = {2p} - 2\n\]\n\nfor every meromorphic \( \left( {1,0}\right) \) -differential \( {\varphi }_{1} \) .\n\nNow let \( {\varphi }_{2} \) be a meromorphic quadratic differential. Then \( {\varphi }_{2}/{\varphi }_{1} \) is a \( \left( {1,0}\right) \) -differential. From \( \deg {D}_{{\varphi }_{1}} + \deg {D}_{{\varphi }_{2}/{\varphi }_{1}} = \deg {D}_{{\varphi }_{2}} \) it thus follows that\n\n\[ \n\deg {D}_{{\varphi }_{2}} = {4p} - 4\n\]\n\n(5.7)\n\nIn particular, every non-zero holomorphic quadratic differential on a Riemann surface of genus \( p \) has \( {4p} - 4 \) zeros.
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No
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Theorem 5.5. On a compact Riemann surface of genus \( p \), the space of holomorphic quadratic differentials has dimension 1 if \( p = 1 \) and \( {3p} - 3 \) if \( p > 1 \) .
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Proof. In the case \( p = 1 \), the Riemann-Roch theorem is not needed to determine the dimension of \( Q \) . We saw in 4.1 that cover transformations are translations \( z \rightarrow z + m{\omega }_{1} + n{\omega }_{2}, m, n \in \mathbb{Z} \) . Formula (3.5) shows, therefore, that \( \varphi \) is a holomorphic quadratic differential for the covering group \( G \) if and only if \( \varphi \left( {z + m{\omega }_{1} + n{\omega }_{2}}\right) = \varphi \left( z\right) \) for all \( m \) and \( n \) . It follows that \( \varphi \in Q \) is a bounded holomorphic function in the complex plane and hence a constant. Conversely, every constant is a quadratic differential for \( G \) . We see that \( \dim Q = 1 \) . Next suppose that \( p > 1 \) . We fix a holomorphic quadratic differential and denote its divisor by \( {D}_{2} \) . After this, we choose \( D = - {D}_{2} \) in (5.6). Then, by (5.4) and (5.7), \[ \dim Q = \dim \left( {{D}_{2} - {D}_{1}}\right) + {3p} - 3. \] (5.8) Now \( \deg \left( {{D}_{2} - {D}_{1}}\right) = \deg {D}_{2} - \deg {D}_{1} = {2p} - 2 > 0 \) . Hence the desired result \( \dim Q = {3p} - 3 \) follows from (5.8) and (5.3).
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Yes
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Theorem 7.1. Every point of a Riemann surface has a neighborhood in which any two points can be joined by a unique shortest curve.
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Proof. Let a point \( p \in S \) be given and suppose first that \( p \) is regular. Let \( V \) be the maximal disc around \( p \) and \( \{ w\left| \right| w \mid < r\} \) its image under a natural parameter \( w = \Phi \left( z\right) \) . Let \( {V}_{0} \subset V \) be the preimage of \( \left| w\right| < r/2 \), and \( {p}_{1},{p}_{2} \) arbitrary points of \( {V}_{0} \) . Then the preimage \( {\gamma }_{0} \) of the line segment connecting \( \Phi \left( {p}_{1}\right) \) and \( \Phi \left( {p}_{2}\right) \) is the unique shortest curve which joins \( {p}_{1} \) and \( {p}_{2} \) on \( S \) . For let \( \gamma \left( { \neq {\gamma }_{0}}\right) \) be an arbitrary curve on \( S \) which joins \( {p}_{1} \) and \( {p}_{2} \) . If \( \gamma \) stays in \( V \) , then clearly \( l\left( {\gamma }_{0}\right) < l\left( \gamma \right) \) . If \( \gamma \) leaves \( V \), then \( l\left( \gamma \right) \geq r > l\left( {\gamma }_{0}\right) \) .\n\nSuppose next that \( p \) is a zero of \( \varphi \) of order \( n \) . We proved in 6.1 that if \( \zeta \) is a natural parameter near \( p \), then\n\n\[ \varphi \left( \zeta \right) = {\left( \frac{n + 2}{2}\right) }^{2}{\zeta }^{n},\;w = \Phi \left( \zeta \right) = {\zeta }^{\left( {n + 2}\right) /2}, \]\n\n(7.2)\n\nin a disc \( \left| \zeta \right| < r \) . Let \( {V}_{0} \) now be the preimage of the disc \( \left| \zeta \right| < {2}^{-2/\left( {n + 2}\right) }r \) on \( S \) . Then any two points \( {p}_{1} \) and \( {p}_{2} \) in \( {V}_{0} \) can be connected by a unique shortest curve. This is either a straight line segment in the \( w \) -plane, or it is composed of two radii in the \( \zeta \) -plane which emanate from the origin. The former case occurs if and only if \( \left| {\arg {\zeta }_{1} - \arg {\zeta }_{2}}\right| < {2\pi }/\left( {n + 2}\right) \), where \( {\zeta }_{1} \) and \( {\zeta }_{2} \) are the \( \zeta \) -images of \( {p}_{1} \) and \( {p}_{2} \) . These conclusions can be drawn from (7.2); for the details we refer to Strebel [6], p. 35.\n\nIt follows from the above that if the shortest curve is the union of two radii, both angles \( \theta \) between these rays satisfy the inequality\n\n\[ \theta \geq \frac{2\pi }{n + 2}. \]\n\n(7.3)\n\nThis \
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Yes
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Lemma 7.1 (Teichmüller’s Lemma). Let \( \varphi \) be holomorphic in the closure of a domain \( A \) in the complex plane which is bounded by a simple closed polygon in the \( \varphi \) -metric, whose sides \( {\gamma }_{j} \) form the angles \( {\theta }_{j} \) at the vertices. If \( {m}_{i} \) and \( {n}_{j} \) denote the orders of the zeros of \( \varphi \) in \( A \) and on \( \partial A \), respectively, then\n\n\[ \sum \left( {1 - \left( {{n}_{j} + 2}\right) \frac{{\theta }_{j}}{2\pi }}\right) = 2 + \sum {m}_{i} \]
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Proof. On \( {\gamma }_{j} \) we have \( \arg \left( {\varphi \left( z\right) d{z}^{2}}\right) = \) constant, and so\n\n\[ d\arg \varphi \left( z\right) + {2d}\left( {\arg {dz}}\right) = 0. \]\n\nThe argument of the tangent vector \( {dz} \) increases by \( {2\pi } - \sum \left( {\pi - {\theta }_{j}}\right) \) after a full turn along \( \partial A \) . This observation, coupled with (7.5) and the Argument principle, yields\n\n\[ {2\pi }\sum {m}_{i} + \sum {\theta }_{j}{n}_{j} = - {4\pi } + 2\sum \left( {\pi - {\theta }_{j}}\right) \]\n\nwhich is (7.4).\n\nIt follows from (7.4) that\n\n\[ \sum \left( {1 - \left( {{n}_{j} + 2}\right) \frac{{\theta }_{j}}{2\pi }}\right) \geq 2 \]\n\n(7.6)\n\nWe conclude that there are at least three angles \( {\theta }_{j} \) so small that\n\n\[ {\theta }_{j} < \frac{2\pi }{{n}_{j} + 2} \]\n\n(7.7)\n\nHence, these angles do not satisfy the angle condition (7.3).
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Yes
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Lemma 7.2. Let \( S = G \) be a simply connected domain in the complex plane and \( {z}_{1} \) and \( {z}_{2} \) points of \( G \) . Then there exists at most one geodesic from \( {z}_{1} \) to \( {z}_{2} \) .
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Proof. Let us assume that there are two geodesics joining \( {z}_{1} \) and \( {z}_{2} \) in, \( G \) . If they do not coincide we can find two subarcs, both from a point \( a \) to a point \( b \), which form a simple closed polygon. The angle condition (7.3) is satisfied at the vertices, except possibly at the two points \( a \) and \( b \) . This is in contradiction with the fact that (7.7) holds for at least three angles.
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Yes
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Lemma 7.3. In a simply connected subdomain of the complex plane every maximal geodesic is a cross-cut.
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Proof. Let \( \gamma \) be a maximal geodesic in a simply connected plane domain \( S = G \) . Fix a point \( {z}_{0} \in \gamma \) and represent a ray of \( \gamma \) with the initial point \( {z}_{0} \) by using its arclength \( u \) as parameter, \( 0 \leq u < {u}_{\infty } \) . Assume that \( \gamma \left( u\right) \) does not tend to \( \partial G \) as \( u \rightarrow {u}_{\infty } \) . Then there is a sequence of points \( {u}_{n} \rightarrow {u}_{\infty } \) such that \( {z}_{n} = \gamma \left( {u}_{n}\right) \rightarrow z \in G \) . By Theorem 7.1, there is a disc \( U \) around \( z \) in which any two points can be joined by a unique shortest curve. Consider the maximal subarc \( {\gamma }_{k} \) of \( \gamma \) which contains the point \( {z}_{k} \) and lies in \( U \) . There is a point \( {z}_{n} \in U, n > k \), which is not on \( {\gamma }_{k} \) . Otherwise \( \gamma \) would terminate at \( z \), which contradicts the fact that every geodesic arc in \( U \) can be continued to \( \partial U \) . Therefore, the part of \( \gamma \) from \( {z}_{k} \) to \( {z}_{n} \) is a geodesic which leaves \( U \) . On the other hand, there is a shortest curve and hence a geodesic from \( {z}_{k} \) to \( {z}_{n} \) inside \( U \) . This is in contradiction with Lemma 7.2.
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Yes
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Theorem 7.3. Let \( S \) be a compact Riemann surface and \( p \) and \( q \) points of \( S \). Then each homotopy class of curves joining \( p \) and \( q \) on \( S \) contains a unique shortest (hence geodesic) arc.
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Proof. As in the proof of Theorem 7.2, we may replace \( S \) by its universal covering surface \( D \). Let two points \( {z}_{1} \) and \( {z}_{2} \) of \( D \) be given. Since the distance from \( {z}_{1} \) and \( {z}_{2} \) to \( \partial D \) is infinite, we can find a Jordan domain \( G,\bar{G} \subset D \), such that \( {z}_{1},{z}_{2} \in G \) and that any arc connecting \( {z}_{1} \) and \( {z}_{2} \) in \( D \) and leaving \( G \) cannot be length minimizing. If \( a \) denotes the infimum of the lengths of the curves in \( D \) which join \( {z}_{1} \) and \( {z}_{2} \), we then obtain the same infimum \( a \) if we restrict attention only to curves which lie in \( G \). Let \( \left( {\gamma }_{i}\right) \) be a minimal sequence of curves in \( G \) from \( {z}_{1} \) to \( {z}_{2} \), i.e., \( l\left( {\gamma }_{i}\right) \rightarrow a \). Subdivide the parameter interval \( \left\lbrack {0, l\left( {\gamma }_{i}\right) }\right\rbrack \) into \( n \) equal parts and take \( n \) so large that the endpoints of the resulting subarcs of \( {\gamma }_{i} \) can be joined by a unique shortest arc in \( D \). That this is possible follows from Theorem 7.1, combined with a standard compactness argument. For a subsequence \( \left( {\gamma }_{{i}_{k}}\right) \), these \( n + 1 \) endpoints converge. By joining the limit points with shortest arcs in \( D \) we obtain a shortest \( \operatorname{arc}\gamma \) from \( {z}_{1} \) to \( {z}_{2} \). Being globally shortest \( \gamma \) is also locally shortest, i.e., a geodesic. The uniqueness follows from Theorem 7.2.
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Yes
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Lemma 7.4. Let \( S \) be a compact Riemann surface, \( f : S \rightarrow S \) a homeomorphism homotopic to the identity, and \( \alpha \) a horizontal arc. Then there is a constant \( M \) , which does not depend on \( \alpha \), such that\n\n\[ l\left( {f\left( \alpha \right) }\right) \geq l\left( \alpha \right) - {2M} \]
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Proof. Let \( h : S \times \left\lbrack {0,1}\right\rbrack \rightarrow S \) be a homotopy from the identity mapping to \( f \) . Fix a point \( p \in S \) and denote by \( {\widetilde{\gamma }}_{p} \) the path \( t \rightarrow h\left( {p, t}\right) \) . Let \( {\gamma }_{p} \) be the (unique) geodesic in the homotopy class of \( {\widetilde{\gamma }}_{p} \) . If \( {p}^{\prime } \) is close to \( p \), the difference \( \left| {l\left( {\gamma }_{p}\right) - l\left( {\gamma }_{{p}^{\prime }}\right) }\right| \) is majorized by the sum of the distances between \( p \) and \( {p}^{\prime } \) and \( f\left( p\right) \) and \( f\left( {p}^{\prime }\right) \) . Hence, the function \( p \rightarrow l\left( {\gamma }_{p}\right) \) is continuous. Since \( S \) is compact, it follows that\n\n\[ M = \mathop{\max }\limits_{{p \in S}}l\left( {\gamma }_{p}\right) < \infty . \]\n\nNow let \( p \) be the initial point and \( q \) the terminal point of the horizontal arc \( \alpha \) . If \( {\gamma }_{q}^{-1} \) denotes the path \( t \rightarrow {\gamma }_{q}\left( {1 - t}\right) \), then \( {\gamma }_{p}f\left( \alpha \right) {\gamma }_{q}^{-1} \) is homotopic to \( \alpha \) . By Theorem 7.2, the geodesic \( \alpha \) is shortest in its homotopy class. Therefore,\n\n\[ l\left( \alpha \right) \leq l\left( {f\left( \alpha \right) }\right) + {2M} \]\n\nas we wished to show.
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Yes
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Theorem 1.1. Let \( \mu \) be a Beltrami differential on a Riemann surface \( S \) . Then there is a quasiconformal mapping of \( S \) onto another Riemann surface with complex dilatation \( \mu \) . The mapping is uniquely determined up to a conformal mapping.
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Proof. We consider \( \mu \) as a Beltrami differential for the covering group \( G \) of \( D \) over \( S \) . By Theorem I.4.4, there is a quasiconformal mapping \( f : D \rightarrow D \) with complex dilatation \( \mu \) . Since (1.1) holds, \( f \) and \( f \circ g \) have the same complex dilatation for every \( g \in G \) . Then \( f \circ g \circ {f}^{-1} \) is conformal, and we conclude that \( f \) induces an isomorphism of \( G \) onto the Fuchsian group \( {G}^{\prime } = \left\{ {f \circ g \circ {f}^{-1} \mid g \in }\right. \) \( G\} \) . If \( \pi \) and \( {\pi }^{\prime } \) denote the canonical projections of \( D \) onto \( S \) and \( {S}^{\prime } = D/{G}^{\prime } \) , then \( \varphi \circ \pi = {\pi }^{\prime } \circ f \) defines a quasiconformal mapping \( \varphi \) of \( S \) onto \( {S}^{\prime } \) . This mapping has the complex dilatation \( \mu \) .\n\nLet \( \psi \) be another quasiconformal mapping of \( S \) with complex dilatation \( \mu \) and \( w : D \rightarrow D \) its lift. Then \( w \circ {f}^{-1} : D \rightarrow D \) is conformal, and so its projection \( \psi \circ {\varphi }^{-1} \) is also conformal.
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Yes
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Theorem 1.2. Let \( S \) and \( {S}^{\prime } \) be Riemann surfaces with non-elementary covering groups \( G \) and \( {G}^{\prime },{\varphi }_{i} : S \rightarrow {S}^{\prime }, i = 0,1 \), two quasiconformal mappings, and \( {f}_{0} \) a lift of \( {\varphi }_{0} \) . Then \( {\varphi }_{0} \) and \( {\varphi }_{1} \) induce the same group isomorphism between \( G \) and \( {G}^{\prime } \) if and only if there is a lift \( {f}_{1} \) of \( {\varphi }_{1} \) which agrees with \( {f}_{0} \) on the limit set of \( G \) .
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Proof. Suppose first that there is a lift \( {f}_{1} \) of \( {\varphi }_{1} \) such that \( {f}_{1} = {f}_{0} \) on the limit set \( L \) of \( G \) . Because \( {f}_{0} \) and \( {f}_{1} \) map \( L \) onto the limit set \( {L}^{\prime } \) of \( {G}^{\prime } \) and because \( L \) is invariant under \( G \), we then have\n\n\[ \n{f}_{0} \circ g \circ {f}_{0}^{-1} = {f}_{1} \circ g \circ {f}_{1}^{-1},\;g \in G, \]\n\n(1.2)\n\nat every point of \( {L}^{\prime } \) . Both sides are Möbius transformations. Since they are equal on a set with at least three points, they agree everywhere.\n\nIn order to prove the necessity of the condition, we now assume that (1.2) is true in \( D \) . Setting \( h = {f}_{0}^{-1} \circ {f}_{1} \), we rewrite (1.2) in the form\n\n\[ \ng \circ h = h \circ g. \]\n\nIf \( z \) is a fixed point of some \( g \), then \( g\left( {h\left( z\right) }\right) = h\left( z\right) \), i.e., \( h\left( z\right) \) is also a fixed point of \( g \) . If \( z \) is an attractive fixed point and \( \zeta \in D \), then for the \( n \) th iterate \( {g}_{n} \) of \( g \) , \( {g}_{n}\left( {h\left( \zeta \right) }\right) \rightarrow z \) as \( n \rightarrow \infty \) . On the other hand, \( {g}_{n}\left( {h\left( \zeta \right) }\right) = h\left( {{g}_{n}\left( \zeta \right) }\right) \rightarrow h\left( z\right) \) . Hence \( h\left( z\right) = z \) for all fixed points of \( G \) . Since these fixed points comprise a dense subset of \( L \) (see IV.4.5), it follows that \( {f}_{0}\left( z\right) = {f}_{1}\left( z\right) \) for all \( z \) in \( L \) .
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Yes
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Theorem 1.3. Let \( S \) be a Riemann surface with a non-elementary covering group. If \( f : S \rightarrow S \) is a conformal mapping homotopic to the identity, then \( f \) is the identity mapping.
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Proof. By Theorem IV.3.5, \( f \) and the identity mapping of \( S \) induce the same group isomorphism of the covering group of \( D \) over \( S \) . By Theorem 1.2, \( f \) has a lift which is the identity mapping of \( D \) . Hence, the projection \( f \) itself is the identity mapping.
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Yes
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Theorem 1.4. Two quasiconformal mappings \( {\varphi }_{i} : S \rightarrow {S}^{\prime }, i = 0,1 \), are homotopic modulo the boundary if and only if they can be lifted to mappings of \( D \) which agree on the boundary.
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Proof. Assume first that \( {\varphi }_{0} \) and \( {\varphi }_{1} \) are homotopic modulo the boundary. If \( {f}_{0} \) is a lift of \( {\varphi }_{0} \), then the lift \( {f}_{1} \) of \( {\varphi }_{1} \) homotopic to \( {f}_{0} \) through the lifted homotopy agrees with \( {f}_{0} \) on the set \( B \) . The mappings \( {f}_{0} \) and \( {f}_{1} \) determine the same group isomorphism (Theorem IV.3.5). From the proof of Theorem 1.2 it follows that \( {f}_{0} = {f}_{1} \) on \( L \) .\n\nConversely, if \( {f}_{0} = {f}_{1} \) on the boundary of \( D \), we construct a homotopy \( {f}_{t} \) from \( {f}_{0} \) to \( {f}_{1} \) as in the proof of Theorem IV.3.5, and conclude again that it can be projected to produce a homotopy from \( {\varphi }_{0} \) to \( {\varphi }_{1} \) . Since \( {f}_{t} \) keeps every point of \( B \) fixed, the projected homotopy is constant on the border of \( S \) .
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Yes
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Theorem 1.5. Let \( S \) and \( {S}^{\prime } \) be compact, topologically equivalent Riemann surfaces. Then every homotopy class of sense-preserving homeomorphisms of \( S \) onto \( {S}^{\prime } \) contains a quasiconformal mapping.
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Proof. Let \( f : S \rightarrow {S}^{\prime } \) be a sense-preserving homeomorphism. Since \( S \) is compact, it has a finite covering by domains \( {U}_{1},{U}_{2},\ldots ,{U}_{n} \), such that \( {U}_{k} \) is conformally equivalent to the unit disc and \( \partial {U}_{k} \) is an analytic curve. Set \( {f}_{0} = f \), and define inductively a sequence of mappings \( {f}_{k}, k = 1,2,\ldots, n \), as follows: \( {f}_{k} = {f}_{k - 1} \) in \( S \smallsetminus {U}_{k} \), while in \( {U}_{k} \), the mapping \( {f}_{k} \) is the Beurling-Ahlfors extension of the boundary values \( {f}_{k - 1} \mid \partial {U}_{k} \) . More precisely, we map \( {U}_{k} \) and \( {f}_{k - 1}\left( {U}_{k}\right) \) conformally onto the upper half-plane \( H \) . Since \( {U}_{k} \) and \( {f}_{k - 1}\left( {U}_{k}\right) \) are Jordan domains, these conformal transformations of \( {U}_{k} \) and \( {f}_{k - 1}\left( {U}_{k}\right) \) onto \( H \) have homeomorphic extensions to the boundary (see I.1.2). We normalize the mappings so that the induced self-mapping \( w \) of \( H \) keeps \( \infty \) fixed. After that, we form the Beurling-Ahlfors extension of \( w \mid \mathbb{R} \) as in I.5.3. By transferring this extension to \( S \) we obtain \( {f}_{k} \mid {U}_{k} \) . The mapping \( {f}_{k} \mid {U}_{k} \) is a diffeomorphism and hence locally quasiconformal. Moreover, if \( {f}_{k - 1} \) is quasiconformal at a point \( z \in \partial {U}_{k} \), then \( {f}_{k} \mid {U}_{k} \cap V \) is quasiconformal for some neighborhood \( V \) of \( z \) (cf. [LV], pp. 84-85). Hence, \( {f}_{k} \) is quasiconformal at \( z \), because \( \partial {U}_{k} \) is a removable singularity (cf. Lemma I.6.1). It follows that \( {f}_{n} \) is a quasiconformal mapping of \( S \), since \( S \) is compact.\n\nThe mapping \( \left( {p, t}\right) \rightarrow t{f}_{k}\left( p\right) + \left( {1 - t}\right) {f}_{k - 1}\left( p\right) \) is a homotopy between \( {f}_{k - 1} \) and \( {f}_{k} \) . It follows that \( {f}_{n} \) is homotopic to \( f \) .
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Yes
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Theorem 2.1. Let \( {f}_{0} : S \rightarrow {S}^{\prime } \) be a quasiconformal mapping and \( F \) the class of all quasiconformal mappings of \( S \) onto \( {S}^{\prime } \) homotopic to \( {f}_{0} \) . Then \( F \) contains an extremal mapping, i.e., one with smallest maximal dilatation.
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Proof. Let \( D \) be a universal covering surface of \( S \) . The theorem is trivial if \( D \) is the extended plane or if \( D \) is the complex plane and \( S \) is non-compact. In the case where \( D \) is the complex plane and \( S \) is compact, the theorem will be proved in 6.4. Hence, we may assume that \( D = H \) is the upper half-plane (cf. IV.4.1).\n\nBy Theorem 1.4, we can lift each \( f \in F \) to a self-mapping \( {w}_{f} \) of \( H \) such that all mappings \( {w}_{f} \) agree on the real axis. The class \( W = \left\{ {{w}_{f} \mid f \in F}\right\} \) contains its quasiconformal limits. Hence, there exists a mapping \( w \in W \) with smallest maximal dilatation (cf. I.5.7). The projection of \( w \) is the extremal sought in \( F \) .
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No
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Theorem 2.2. The Teichmüller space \( {T}_{S} \) is pathwise connected.
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Proof. The geodesic \( t \rightarrow \left\lbrack {\mu }_{t}\right\rbrack \) is a path joining the origin to the point \( p \) in \( {T}_{s} \) ; the path \( t \rightarrow \left\lbrack {t\mu }\right\rbrack \) of \( {T}_{S} \) also has this property.
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No
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Theorem 2.3. The conformal structures \( {H}_{1} \) and \( {H}_{2} \) induced by the Beltrami differentials \( {\mu }_{1} \) and \( {\mu }_{2} \) on the Riemann surface \( S \) are deformation equivalent if and only if \( {\mu }_{1} \) and \( {\mu }_{2} \) determine the same point in the Teichmüller space \( {T}_{S} \) .
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Proof. Let \( {f}_{i}, i = 1,2 \), be quasiconformal mappings of \( S \) with complex dilatations \( {\mu }_{i} \) . If \( \varphi : \left( {S,{H}_{1}}\right) \rightarrow \left( {S,{H}_{2}}\right) \) is a conformal mapping homotopic to the identity, we first conclude that the mapping\n\n\[ h = {f}_{2} \circ \varphi \circ {f}_{1}^{-1} : {f}_{1}\left( S\right) \rightarrow {f}_{2}\left( S\right) \]\n\nis conformal. Also, we see that \( {f}_{2} \circ {f}_{1}^{-1} \) is homotopic to \( h \) . It follows that \( {\mu }_{1} \) and \( {\mu }_{2} \) are equivalent.\n\nConversely, if \( {\mu }_{1} \) and \( {\mu }_{2} \) are equivalent, there is a conformal map \( h : {f}_{1}\left( S\right) \rightarrow \) \( {f}_{2}\left( S\right) \) such that \( \varphi = {f}_{2}^{-1} \circ h \circ {f}_{1} : S \rightarrow S \) is homotopic to the identity. In addition, \( \varphi : \left( {S,{H}_{1}}\right) \rightarrow \left( {S,{H}_{2}}\right) \) is conformal, and so \( {H}_{1} \) is equivalent to \( {H}_{2} \) .\n\nWe conclude that the Teichmüller space \( {T}_{S} \) can be characterized as the set of equivalence classes of conformal structures \( {H}_{\mu } \) on \( S \) modulo deformation. Note that a conformal structure \( {H}^{\prime } \) on \( S \) is of the form \( {H}_{\mu } \) if and only if id: \( \left( {S, H}\right) \rightarrow \) \( \left( {S,{H}^{\prime }}\right) \) is quasiconformal.
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Yes
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Theorem 2.4. On a compact Riemann surface \( S \), every conformal structure is deformation equivalent to a structure induced by a Beltrami differential of S.
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Proof. Let \( H \) be the given and \( {H}^{\prime } \) an arbitrary conformal structure on \( S \) . By Theorem 1.5, there is a quasiconformal mapping \( f : \left( {S, H}\right) \rightarrow \left( {S,{H}^{\prime }}\right) \) which is homotopic to the identity. Let \( f \) have the complex dilatation \( \mu \) . Then \( {H}^{\prime } = \) \( {f}_{ * }\left( {H}_{\mu }\right) \) (formula (2.5)). But \( f : \left( {S,{H}_{\mu }}\right) \rightarrow \left( {S,{f}_{ * }\left( {H}_{\mu }\right) }\right) \) is a conformal mapping homotopic to the identity. Consequently, \( {H}^{\prime } = {f}_{ * }\left( {H}_{\mu }\right) \) is deformation equivalent to \( {H}_{\mu } \) .
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Yes
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Theorem 2.5. The Teichmüller space of a compact Riemann surface is isomorphic to the set of equivalence classes of conformal structures modulo deformation.
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This result can also be expressed in somewhat different terms. Let \( \mathcal{H}\left( S\right) \) denote the set of all conformal structures of \( S \) . The group Homeo \( {}^{ + }\left( S\right) \) consisting of all sense-preserving homeomorphic self-mappings of \( S \) acts on \( \mathcal{H}\left( S\right) \) : If \( H \in \mathcal{H}\left( S\right) \) and \( f \in {\operatorname{Homeo}}^{ + }\left( S\right) \), then \( {f}_{ * }\left( H\right) \in \mathcal{H}\left( S\right) \) . Let \( {\operatorname{Homeo}}_{0}\left( S\right) \) be the subgroup of \( {\operatorname{Homeo}}^{ + }\left( S\right) \) whose mappings are homotopic to the identity. Then \( H,{H}^{\prime } \in \mathcal{H}\left( S\right) \) are deformation equivalent if and only if there is an \( f \in {\operatorname{Homeo}}_{0}\left( S\right) \) such that \( {f}_{ * }\left( H\right) = {H}^{\prime } \) . It follows, therefore, that for a compact surface \( S \) we have the isomorphism \[ {T}_{S} \simeq \mathcal{H}\left( S\right) /{\operatorname{Homeo}}_{0}\left( S\right) \]
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Yes
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Theorem 2.6. The Teichmüller spaces of two quasiconformally equivalent Riemann surfaces are isometrically bijective.
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Proof. Let \( S \) and \( {S}^{\prime } \) be Riemann surfaces and \( h \) a quasiconformal mapping of \( S \) onto \( {S}^{\prime } \) . The mapping \( f \rightarrow f \circ {h}^{-1} \) is a bijection of the family of all quasiconformal mappings \( f \) of \( S \) onto the family of all quasiconformal mappings of \( {S}^{\prime } \) . If \( {w}_{i} = {f}_{i} \circ {h}^{-1} \), we have \( {w}_{2} \circ {w}_{1}^{-1} = {f}_{2} \circ {f}_{1}^{-1} \) . We first conclude that \( {f}_{1} \) and \( {f}_{2} \) determine the same point of \( {T}_{S} \) if and only if \( {w}_{1} \) and \( {w}_{2} \) determine the same point in \( {T}_{{S}^{\prime }} \), i.e.,\n\n\[ \left\lbrack f\right\rbrack \rightarrow \left\lbrack {f \circ {h}^{-1}}\right\rbrack \]\n\n(2.6)\n\nis a bijective mapping of \( {T}_{S} \) onto \( {T}_{{S}^{\prime }} \) . It also follows that (2.6) is an isometry, i.e., it leaves all Teichmüller distances invariant.\n\nUnder (2.6) the point \( \left\lbrack h\right\rbrack \) of \( {T}_{S} \) is mapped to the origin of \( {T}_{{S}^{\prime }} \) . We shall later utilize this simple method of moving an arbitrary point of one Teichmüller space to the origin of another isometric Teichmüller space.
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Yes
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Theorem 2.7. The Riemann space is the quotient of the Teichmüller space by the modular group.
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Proof. Assume first that the points \( \left\lbrack f\right\rbrack \) and \( \left\lbrack g\right\rbrack \) of \( {T}_{S} \) are equivalent under \( \operatorname{Mod}\left( S\right) \) . We then have a quasiconformal mapping \( h : S \rightarrow S \) such that \( f \circ {h}^{-1} \) is equivalent to \( g \) . But this means that there is a conformal mapping of \( f\left( S\right) \) onto \( g\left( S\right) \), i.e., \( f \) and \( g \) determine the same point of \( {R}_{S} \) .\n\nConversely, let \( f \) and \( g \) represent the same point of \( {R}_{S} \) . Then a conformal mapping \( \varphi : f\left( S\right) \rightarrow g\left( S\right) \) exists, and \( h = {g}^{-1} \circ \varphi \circ f \) is a quasiconformal self-mapping of \( S \) . From \( g = \varphi \circ \left( {f \circ {h}^{-1}}\right) \) we see that \( g \) and \( f \circ {h}^{-1} \) determine the same point of \( {T}_{S} \) .
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Yes
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Theorem 3.1. The Beltrami differentials \( \mu \) and \( v \) of \( S \) are equivalent if and only if \( {f}^{\mu }\left| {\mathbb{R} = {f}^{v}}\right| \mathbb{R} \) or if and only if \( {f}_{\mu }\left| {H = {f}_{v}}\right| H \) .
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Proof. Let us first assume that \( \mu \) and \( v \) are equivalent. Let \( \varphi \) and \( \psi \) be quasiconformal mappings of \( S \) which lift to \( {f}^{\mu } \) and \( {f}^{v} \), respectively. Then there is a conformal map \( \eta : \varphi \left( S\right) \rightarrow \psi \left( S\right) \) such that \( \eta \circ \varphi \) is homotopic to \( \psi \) . By Theorem 1.4, we have \( {f}^{v} = h \circ {f}^{\mu } \) on the real axis \( \mathbb{R} \), where \( h \), as a lift of \( \eta \), is a Möbius transformation. Since \( {f}^{\mu } \) and \( {f}^{v} \) both fix \( 0,1,\infty \), it follows that \( h \) is the identity.\n\nSuppose, conversely, that \( {f}^{\mu } = {f}^{v} \) on the boundary \( \mathbb{R} \) . Then \( {f}^{\mu } \) and \( {f}^{v} \) induce the same isomorphism of the covering group of \( {H}^{\prime } \) over \( S \) onto a Fuchsian group \( {G}^{\prime } \) . The projections of \( {f}^{\mu } \) and \( {f}^{v} \) map \( S \) onto the same Riemann surface \( {H}^{\prime }/{G}^{\prime } \), and by Theorem 1.4, these projections are homotopic. It follows that \( \mu \) and \( v \) are equivalent.
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Yes
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Lemma 3.1. Let \( \left\lbrack {\mu }_{n}\right\rbrack \rightarrow \left\lbrack \mu \right\rbrack \) in \( {T}_{S},{\begin{Vmatrix}{\mu }_{n}\end{Vmatrix}}_{\infty } \leq k < 1 \), and \( {\mu }_{n} \rightarrow v \) a.e. Then \( \left\lbrack \mu \right\rbrack = \) \( \left\lbrack v\right\rbrack \) in \( {T}_{S} \) .
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Proof. Let \( {\lambda }_{n} \in \left\lbrack {\mu }_{n}\right\rbrack \) be an extremal complex dilatation for which \( {\tau }_{S}\left( \left\lbrack {\mu }_{n}\right\rbrack \right. \) , \( \left. \left\lbrack \mu \right\rbrack \right) = \operatorname{artanh}{\begin{Vmatrix}\left( {\lambda }_{n} - \mu \right) /\left( 1 - \bar{\mu }{\lambda }_{n}\right) \end{Vmatrix}}_{\infty } \) (formula (2.3)). The hypothesis \( \left\lbrack {\mu }_{n}\right\rbrack \rightarrow \) \( \left\lbrack \mu \right\rbrack \) then implies that \( {\lambda }_{n} \rightarrow \mu \) in \( {L}^{\infty } \) . By Theorem I.4.6, \( {f}_{{\mu }_{n}} \rightarrow {f}_{v} \) and \( {f}_{{\lambda }_{n}} \rightarrow {f}_{\mu } \) . Since \( {f}_{{\mu }_{n}}\left| {H = {f}_{{\lambda }_{n}}}\right| H \) it follows that \( {f}_{\mu }\left| {H = {f}_{v}}\right| H \), and so \( \left\lbrack \mu \right\rbrack = \left\lbrack v\right\rbrack \) .
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Yes
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Theorem 3.3. The mapping \( g \rightarrow {f}_{\mu } \circ g \circ {f}_{\mu }^{-1} \) defines an isomorphism of the covering group \( G \) onto a group \( {G}_{\mu } \) of Möbius transformations acting on the quasidisc \( {f}_{\mu }\left( {H}^{\prime }\right) \) .
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Proof. Consider the quasiconformal mapping \( {f}_{\mu } \circ g \circ {f}_{\mu }^{-1}, g \in G \), of the plane. It is conformal in \( {f}_{\mu }\left( H\right) \), because \( {f}_{\mu } \mid H \) is conformal. Since \( \mu \) is a Beltrami differential for \( G \), the mappings \( {f}_{\mu } \) and \( {f}_{\mu } \circ g \) have the same complex dilatation. It follows that \( {f}_{\mu } \circ g \circ {f}_{\mu }^{-1} \) is conformal in \( {f}_{\mu }\left( {H}^{\prime }\right) \) also. The common boundary of \( {f}_{\mu }\left( H\right) \) and \( {f}_{\mu }\left( {H}^{\prime }\right) \), being the image of the real axis under \( {f}_{\mu } \), is a quasicircle. We conclude, therefore, from Lemma I.6.1 that \( {f}_{\mu } \circ g \circ {f}_{\mu }^{-1} \) is a Möbius transformation.\n\nBy the terminology we adopted in IV.4.6, the group\n\n\[ \n{G}_{\mu } = \left\{ {{f}_{\mu } \circ g \circ {f}_{\mu }^{-1} \mid g \in G}\right\} \n\]\n\nis quasi-Fuchsian. A quasi-Fuchsian group of this special type is called a quasiconformal deformation of the Fuchsian group \( G \) . Such groups were discovered by Bers [4].
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Yes
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Theorem 3.5. The mapping\n\n\\[ \n\\left\\lbrack \\mu \\right\\rbrack \\rightarrow {f}^{\\mu } \\mid \\mathbb{R} \n\\] \n\n(3.4)\n\nis a homeomorphism of \\( \\left( {{T}_{S},{\\tau }_{S}}\\right) \\) onto \\( \\left( {X\\left( G\\right) ,\\rho }\\right) \\) .
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Proof. By Theorem 3.1, the mapping (3.4) is well defined and injective. By Theorem 3.4, it is surjective.\n\nBy Theorem III.3.1, the mapping (3.4) is a homeomorphism of \\( \\left( {T,\\tau }\\right) \\) onto \\( \\left( {X,\\rho }\\right) \\) . Hence (3.4), which maps \\( {T}_{S} \\) bijectively onto \\( X\\left( G\\right) \\), is a homeomorphism of \\( \\left( {{T}_{S},\\tau }\\right) \\) onto \\( \\left( {X\\left( G\\right) ,\\rho }\\right) \\) . From \\( \\tau \\mid {T}_{S} \\leq {\\tau }_{S} \\) we thus conclude that (3.4) is a continuous mapping of \\( \\left( {{T}_{S},{\\tau }_{S}}\\right) \\) onto \\( \\left( {X\\left( G\\right) ,\\rho }\\right) \\) . The proof would be complete if we had an inequality in the opposite direction between \\( \\tau \\mid {T}_{S} \\) and \\( {\\tau }_{S} \\), to demonstrate that these two metrics are topologically equivalent. Such an inequality can be derived, for instance, by means of the right-hand inequality (5.10) in I.5.7. We content ourselves here with this remark, because we shall study the relationships between \\( \\tau \\mid {T}_{S} \\) and \\( {\\tau }_{S} \\) in detail in the next section (Theorem 4.7).
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No
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Lemma 4.1. The following three conditions are equivalent:\n\n\\( {1}^{ \\circ }{S}_{{f}_{\\mu } \\mid H} \\) is a quadratic differential for \\( G \\) ;\n\n\\( {2}^{ \\circ }{f}_{\\mu } \\circ g \\circ {f}_{\\mu }^{-1} \\) agrees with a Möbius transformation in \\( {f}_{\\mu }\\left( H\\right) \\) for \\( g \\in G \\) ;\n\n\\( {3}^{ \\circ }{f}^{\\mu } \\circ g \\circ {\\left( {f}^{\\mu }\\right) }^{-1} \\) agrees with a Möbius transformation on \\( \\mathbb{R} \\) for \\( g \\in G \\) .
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Proof. As we already remarked, the equivalence of \\( {1}^{ \\circ } \\) and \\( {2}^{ \\circ } \\) follows directly from (4.1). If \\( {2}^{ \\circ } \\) holds, i.e., if \\( {f}_{\\mu } \\circ g \\circ {f}_{\\mu }^{-1} = w \\) in \\( {f}_{\\mu }\\left( H\\right) \\), where \\( w \\) is a Möbius transformation, then \\( h = {f}^{\\mu } \\circ {f}_{\\mu }^{-1} \\circ w \\circ {f}_{\\mu } \\circ {\\left( {f}^{\\mu }\\right) }^{-1} : {H}^{\\prime } \\rightarrow {H}^{\\prime } \\) is a Möbius transformation which coincides with \\( {f}^{\\mu } \\circ g \\circ {\\left( {f}^{\\mu }\\right) }^{-1} \\) on \\( \\mathbb{R} \\) . Hence \\( {3}^{ \\circ } \\) follows from \\( {2}^{ \\circ } \\) .\n\nConversely, assume that \\( {3}^{ \\circ } \\) holds, i.e., that \\( {f}^{\\mu } \\circ g \\circ {\\left( {f}^{\\mu }\\right) }^{-1} = h \\) on \\( \\mathbb{R} \\), where \\( h \\) is a Möbius transformation. Set \\( w = {f}_{\\mu } \\circ {\\left( {f}^{\\mu }\\right) }^{-1} \\circ h \\circ {f}^{\\mu } \\circ {f}_{\\mu }^{-1} \\) in the closure of \\( {f}_{\\mu }\\left( {H}^{\\prime }\\right) \\), and \\( w = {f}_{\\mu } \\circ g \\circ {f}_{\\mu }^{-1} \\) in \\( {f}_{\\mu }\\left( H\\right) \\) . Then \\( w \\) is a homeomorphism of the plane and is conformal in \\( {f}_{\\mu }\\left( H\\right) \\) and \\( {f}_{\\mu }\\left( {H}^{\\prime }\\right) \\) . Hence \\( w \\) is a Möbius transformation, and so \\( {2}^{ \\circ } \\) follows from \\( {3}^{ \\circ } \\) .
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Yes
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Theorem 4.1. The Teichmüller spaces satisfy the relation\n\n\[ T\\left( G\\right) = Q\\left( G\\right) \\cap T\\left( 1\\right) \]
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Proof. The inclusion \( T\\left( G\\right) \\subset Q\\left( G\\right) \\cap T\\left( 1\\right) \) follows directly from the definitions. We choose an arbitrary point \( {S}_{f} \\in Q\\left( G\\right) \\cap T\\left( 1\\right) \) and prove that \( {S}_{f} \\in T\\left( G\\right) \) .\n\nLet \( w \) be a conformal mapping of the lower half-plane \( {H}^{\\prime } \) onto the complement of the closure of \( f\\left( H\\right) \), normalized so that \( {w}^{-1}\\left( {f\\left( \\infty \\right) }\\right) = \\infty \) . Since \( {S}_{f} \\in T\\left( 1\\right) \), the boundary of \( f\\left( H\\right) \) is a quasicircle. Hence, the function \( h = {w}^{-1} \\circ f \), defined on the real axis, is quasisymmetric; we may assume that \( h \) is normalized. Furthermore, for \( g \\in G \) ,\n\n\[ h \\circ g = {w}^{-1} \\circ f \\circ g \\circ {f}^{-1} \\circ f = {w}^{-1} \\circ \\left( {f \\circ g \\circ {f}^{-1}}\\right) \\circ w \\circ h. \]\n\nSince \( {S}_{f} \\in U\\left( G\\right) \), the mapping \( f \\circ g \\circ {f}^{-1} \) agrees with a Möbius transformation \( {g}_{1} \) in \( f\\left( H\\right) \) . Then \( {w}^{-1} \\circ {g}_{1} \\circ w \), which maps \( {H}^{\\prime } \) onto itself, agrees with a Möbius transformation \( {g}_{2} \) in \( {H}^{\\prime } \) . It follows from (4.5) that \( h \) induces an isomorphism of \( G \) onto the group \( \\left\\{ {{g}_{2} \\mid g \\in G}\\right\\} \) of Möbius transformations acting on \( {H}^{\\prime } \) . In other words \( h \\in X\\left( G\\right) \).\n\nNext we utilize Theorem 3.4. It follows that there is a quasiconformal extension \( \\varphi \) of \( h \) to the lower half-plane which also fulfills the condition \( \\varphi \\circ g = {g}_{2} \\circ \\varphi \) for every \( g \\in G \) . Then \( {f}_{1} = w \\circ \\varphi \) is a quasiconformal extension of \( f \) to the lower half-plane. For \( g \\in G \) we have\n\n\[ {f}_{1} \\circ g = w \\circ {g}_{2} \\circ \\varphi = {g}_{1} \\circ w \\circ \\varphi = {g}_{1} \\circ {f}_{1}. \]\n\nThis shows that \( {f}_{1} \\circ g \\circ {f}_{1}^{-1} \) agrees with a Möbius transformation in \( {f}_{1}\\left( {H}^{\\prime }\\right) \) , and it follows that \( {S}_{f} \\in T\\left( G\\right) \).
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Yes
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Theorem 4.2. The set \( T\left( G\right) \) is closed in \( T\left( 1\right) \) .
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Proof. The relation (4.4) is equivalent to \( T\left( G\right) = U\left( G\right) \cap T\left( 1\right) \) . Since \( U\left( G\right) \) is closed in \( Q\left( 1\right) \), the theorem follows.
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Yes
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Theorem 4.3. The set \( T\left( G\right) \) is open in \( Q\left( G\right) \) .
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Proof. This can be read from (4.4), since \( T\left( 1\right) \) is open in \( Q\left( 1\right) \) .
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No
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Theorem 4.4. The ball\n\n\[ B\left( {0,2}\right) = \{ \varphi \in Q\left( G\right) \mid \parallel \varphi \parallel < 2\} \]\n\nlies in \( T\left( G\right) \) .
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Proof. In III.4.3 we remarked that \( \{ \varphi \in Q\left( 1\right) \mid \parallel \varphi \parallel < 2\} \) lies in \( T\left( 1\right) \) (Theorem II.5.1). Hence, the theorem follows immediately from (4.4).
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No
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Theorem 4.5. Every point of the Teichmüller space \( {T}_{S} \) can be represented by a real analytic Beltrami differential and by a real analytic quasiconformal mapping.
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Proof. Let a point \( \left\lbrack \mu \right\rbrack = p \in {T}_{S} \) be given. Suppose first that \( p \) can be represented by a quasiconformal mapping whose maximal dilatation is \( < 2 \) . Then \( p \) lies in the set (4.7), and so \( p \) can be represented by \( z \rightarrow - 2{y}^{2}\varphi \left( \bar{z}\right) \), which is a real analytic complex dilatation. We also have an explicit expression for the corresponding mapping \( {f}^{\mu } \) (cf. II.5.1 and II.5.2) from which it becomes apparent that \( {f}^{\mu } \) is real analytic.\n\nThe general case is handled by induction. Assuming that the theorem is true if \( {\tau }_{S}\left( {p,0}\right) < r \), we show that it holds if \( {\tau }_{S}\left( {p,0}\right) < {2r} \) . Let \( {f}^{\mu } \) be an extremal for the point \( p \), with \( {\tau }_{S}\left( {p,0}\right) < {2r} \) . We write \( {f}^{\mu } = {f}^{{\mu }_{1}} \circ {f}^{{\mu }_{2}} \), where \( {\mu }_{2}\left( z\right) \) is the middle point of the line segment from 0 to \( \mu \left( z\right) \) in the non-euclidean metric of the unit disc (cf. Theorem III.2.2). Then \( {\mu }_{2} \) is a Beltrami differential for \( G \) and \( {\mu }_{1} \) a Beltrami differential for \( {G}^{{\mu }_{2}} \) . Moreover, \( {\tau }_{S}\left( {\left\lbrack {\mu }_{2}\right\rbrack ,0}\right) < r \) and \( {\tau }_{{S}^{\prime }}\left( {\left\lbrack {\mu }_{1}\right\rbrack ,0}\right) < \) \( r \) with \( {S}^{\prime } = {H}^{\prime }/{G}^{{\mu }_{2}} \) . From this the theorem follows.
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Yes
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For every \( {s}_{\mu } \in T\left( G\right) \), the ball\n\n\[ B\left( {{s}_{\mu },{\sigma }_{I}\left( {A}_{\mu }\right) }\right) = \left\{ {\varphi \in Q\left( G\right) \mid q\left( {{s}_{\mu },\varphi }\right) < {\sigma }_{I}\left( {A}_{\mu }\right) }\right\} \]\n\nis contained in \( T\left( G\right) \) .
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In III.5.3 we proved that \( \left\{ {\varphi \in Q\left( 1\right) \mid q\left( {{s}_{\mu },\varphi }\right) < {\sigma }_{I}\left( {A}_{\mu }\right) }\right\} \) lies in \( T\left( 1\right) \) . Consequently, the theorem follows immediately from (4.4).
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No
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Theorem 4.8. The mapping\n\n\[ \left\lbrack \mu \right\rbrack \rightarrow {S}_{{f}_{\mu } \mid H} \]\n\n(4.15)\n\nis a homeomorphism of \( \left( {{T}_{S},{\tau }_{S}}\right) \) onto \( \left( {T\left( G\right), q}\right) \) .
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Proof. By Theorem III.4.1, this mapping is a homeomorphism of \( \left( {{T}_{S},\tau \mid {T}_{S}}\right) \) onto \( T\left( G\right) \) . By Theorem 4.7, the metrics \( {\tau }_{S} \) and \( \tau \mid {T}_{S} \) are equivalent, and the theorem follows.
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Yes
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Theorem 5.1. The function\n\n\\[ \n\mu \rightarrow \Lambda \left( \mu \right) = {S}_{{f}_{\mu } \mid H} \n\\]\n\n(5.1)\n\nwhich maps the open unit ball \\( B\\left( G\\right) \\) of the space of measurable \\( \\left( {-1,1}\\right) \\) - differentials for \\( G \\) into the space \\( Q\\left( G\\right) \\) of holomorphic quadratic differentials for \\( G \\), is holomorphic.
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Proof. We have already seen that \\( Q\\left( G\\right) \\) is a Banach space. The ball \\( B\\left( G\\right) \\) is an open subset of the Banach space \\( {L}^{\\infty }\\left( G\\right) \\) of measurable \\( \\left( {-1,1}\\right) \\) -differentials for \\( G \\) with finite \\( {L}^{\\infty } \\) -norm. Fix \\( \\mu, v \\in B\\left( G\\right) \\) .\n\nFor \\( z \\in H \\) and \\( \\varphi \\in Q\\left( G\\right) \\), we set \\( {\\alpha }_{z}\\left( \\varphi \\right) = \\varphi \\left( z\\right) \\) . Then \\( A = \\left\\{ {{\\alpha }_{z} \\mid z \\in H}\\right\\} \\) is a total set in the dual of \\( Q\\left( G\\right) \\) . We apply condition (ii) of Lemma 5.1 to the function\n\n\\[ \nw \\rightarrow s\\left( w\\right) = {S}_{{f}_{\\mu + {wv}} \\mid H} \\]\n\n(5.2)\n\nThe set \\( U \\) in condition (ii) is now the neighborhood \\( \\{ w\\left| \\right| w \\mid < \\left( {1 - \\parallel \\mu {\\parallel }_{\\infty }}\\right) / \\) \\( \\left. {\\parallel v{\\parallel }_{\\infty }}\\right\\} \\) of the origin in the complex plane. Then \\( \\mu + {wv} \\in B\\left( G\\right) \\) . Furthermore, let \\( F = Q\\left( G\\right) \\) and \\( \\alpha = {\\alpha }_{z} \\) .\n\nBy Corollary II.3.1 and the Remark following it, the function\n\n\\[ \nw \\rightarrow {\\alpha }_{z} \\circ s = {S}_{{f}_{\\mu + {wv}} \\mid H}\\left( z\\right) \\]\nis holomorphic in \\( U \\) for every \\( z \\in H \\) . By formula (III.4.4), the function \\( s \\) is continuous in \\( U \\) . Hence, by condition (ii) of Lemma 5.1, the function (5.2) is holomorphic in \\( U \\) . Using this fact we conclude from condition (i) of Lemma 5.1 that (5.1) is holomorphic in \\( B\\left( G\\right) \\) .
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Yes
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Theorem 5.2. The atlas\n\n\\[ \n\\left\\{ {\\left( {{V}_{\\mu },{h}_{\\mu }}\\right) \\mid \\mu \\in B\\left( G\\right) }\\right\\} \n\\]\n\n(5.8)\n\ndefines a complex analytic structure on the Teichmüller space \\( {T}_{S} \\) . The Bers imbedding \\( \\left\\lbrack \\mu \\right\\rbrack \\rightarrow {\\left. {S}_{{f}_{\\mu }}\\right| }_{H} \\) of \\( {T}_{S} \\) into \\( Q\\left( G\\right) \\) is holomorphic with respect to this structure.
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Proof. Assuming that \\( {V}_{\\mu } \\) and \\( {h}_{\\mu } \\) are defined by (5.6) and (5.7), we choose two elements \\( {\\mu }_{1} \\) and \\( {\\mu }_{2} \\) of \\( B\\left( G\\right) \\) such that \\( {V}_{{\\mu }_{1}} \\cap {V}_{{\\mu }_{2}} \\) is not empty. In \\( {h}_{{\\mu }_{1}}\\left( {{V}_{{\\mu }_{1}} \\cap {V}_{{\\mu }_{2}}}\\right) \\) we have\n\n\\[ \n{h}_{{\\mu }_{2}} \\circ {h}_{{\\mu }_{1}}^{-1} = {\\Lambda }_{{\\mu }_{2}} \\circ {\\widetilde{\\alpha }}_{{\\mu }_{2}} \\circ {\\widetilde{\\alpha }}_{{\\mu }_{1}}^{-1} \\circ {\\sigma }_{{\\mu }_{1}}.\n\\]\n\nAll mappings on the right-hand side are holomorphic, as we have seen. Consequently, as a composition of holomorphic functions, \\( {h}_{{\\mu }_{2}} \\circ {h}_{{\\mu }_{1}}^{-1} \\) is holomorphic. By changing the roles of \\( {\\mu }_{1} \\) and \\( {\\mu }_{2} \\) we conclude that \\( {h}_{{\\mu }_{2}} \\circ {h}_{{\\mu }_{1}}^{-1} \\) is biholomorphic. Hence,(5.8) defines a complex analytic structure for \\( {T}_{S} \\) .\n\nIt is not difficult to see that the complex structure we obtained by means of the atlas (5.8) is independent of the representation \\( {H}^{\\prime }/G \\) we used for the Riemann surface \\( S \\) . Theorem 5.5 expresses an even stronger result.\n\nIn order to complete the proof of the theorem, we still have to show that the Bers imbedding \\( \\lambda : {T}_{S} \\rightarrow Q\\left( G\\right) \\) is holomorphic, i.e., that \\( \\lambda \\circ {h}_{\\mu }^{-1} \\) is holomorphic in \\( {B}_{\\mu }\\left( {0,2}\\right) \\) . Now\n\n\\[ \n\\lambda \\circ {h}_{\\mu }^{-1} = \\Lambda \\circ {\\widetilde{\\alpha }}_{\\mu }^{-1} \\circ {\\sigma }_{\\mu }\n\\]\n\nSince all mappings on the right are holomorphic, their composition \\( \\lambda \\circ {h}_{\\mu }^{-1} \\) is holomorphic.
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Yes
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Theorem 5.3. The canonical projection\n\n\\[ \n\\pi : B\\left( G\\right) \\rightarrow {T}_{S}\n\\] \nis holomorphic, and it has local holomorphic sections everywhere in \\( {T}_{s} \\) .
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Proof. First of all, we have\n\n\\[ \n{h}_{\\mu } \\circ \\pi = {\\Lambda }_{\\mu } \\circ {\\widetilde{\\alpha }}_{\\mu }\n\\] \n\nSince \\( {\\Lambda }_{\\mu } \\) and \\( {\\widetilde{\\alpha }}_{\\mu } \\) are holomorphic, it follows that \\( \\pi \\) is holomorphic.\n\nNext, let us consider the mapping\n\n\\[ \n{\\psi }_{\\mu } = {\\widetilde{\\alpha }}_{\\mu }^{-1} \\circ {\\sigma }_{\\mu } \\circ {h}_{\\mu }\n\\] \n\nof \\( {V}_{\\mu } \\) into \\( B\\left( G\\right) \\) . All functions on the right-hand side are holomorphic. Therefore, their composition \\( {\\psi }_{\\mu } \\) is holomorphic. From the definitions we infer that\n\n\\[ \n\\pi \\circ {\\psi }_{\\mu } = \\text{identity mapping of}{V}_{\\mu }\\text{.}\n\\] \n\n(5.9)\n\nConsequently, \\( {\\psi }_{\\mu } \\) is the desired section.\n\nSince \\( {\\psi }_{\\mu }\\left( {\\pi \\left( \\mu \\right) }\\right) = \\mu \\), we conclude that \\( \\pi \\) is an open mapping. We remark that Theorem 5.3 determines the complex analytic structure of \\( {T}_{S} \\) uniquely.
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Yes
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Theorem 5.4. The Bers imbedding \( \lambda : {T}_{S} \rightarrow T\left( G\right) \) is biholomorphic.
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Proof. Suppose first that \( Q\left( G\right) \) is finite dimensional. (By IV.5.5, this is the case if \( S \) is compact; cf. also 9.7.) We then conclude directly from Theorem 5.2 that \( \lambda \) is biholomorphic using the theorem by which a holomorphic bijection is always biholomorphic in finite dimensional manifolds (Narasimhan [1], p. 86).\n\nIn the general case, we fix a point \( {s}_{\mu } \in T\left( G\right) \) and consider Schwarzians \( {s}_{v} \in T\left( G\right) \) lying close to \( {s}_{\mu } \) . Then \( f = {f}_{v} \circ {f}_{\mu }^{-1} \) has a small Schwarzian derivative in the quasidisc \( {f}_{\mu }\left( H\right) \) . One proves that \( f \) has a quasiconformal extension whose complex dilatation is in \( B\left( {G}_{\mu }\right) \) and depends holomorphically on \( {s}_{v} \) (Bers [9], Theorem 6; cf. the remark at the end of II.5.1). The conclusion is that \( \Lambda \) has a local holomorphic section at \( {s}_{\mu } \) . Since \( \Lambda = \lambda \circ \pi \), we deduce from Theorem 5.3 that \( \lambda \) is biholomorphic.
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Yes
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Theorem 5.5. Quasiconformally equivalent Riemann surfaces have isometrically and biholomorphically isomorphic Teichmüller spaces.
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Proof. Let \( S = {H}^{\prime }/G \) and \( {S}^{\prime } = {H}^{\prime }/{G}^{\prime } \) be quasiconformally equivalent Riemann surfaces. We consider a lift of a quasiconformal mapping of \( S \) onto \( {S}^{\prime } \) to a self-mapping of the lower half-plane. There is no loss of generality in assuming that the lift is of the form \( {f}^{\mu } \) . We have \( \mu \in B\left( G\right) \) .\n\nThe mapping \( {\widetilde{\alpha }}_{\mu } \) induced by \( \mu \) is now a biholomorphic mapping of \( B\left( G\right) \) ’ onto \( B\left( {G}^{\prime }\right) \) . It induces the mapping \( {\alpha }_{\mu } \) of \( {T}_{S} \) onto \( {T}_{{S}^{\prime }} \) . The theorem follows when we prove that \( {\alpha }_{\mu } \) is biholomorphic.\n\nIf \( {\pi }^{\prime } : B\left( {G}^{\prime }\right) \rightarrow {T}_{{S}^{\prime }} \) is the canonical projection, we have\n\n\[{\alpha }_{\mu } \circ \pi = {\pi }^{\prime } \circ {\widetilde{\alpha }}_{\mu }\]\n\n(5.10)\n\nConsider an arbitrary element \( v \in B\left( G\right) \) . In view of (5.9) and (5.10), we have in \( {V}_{v} \)\n\n\[{\alpha }_{\mu } = {\alpha }_{\mu } \circ \pi \circ {\psi }_{v} = {\pi }^{\prime } \circ {\widetilde{\alpha }}_{\mu } \circ {\psi }_{v}.\n\]\n\nAgain, on the right all functions are holomorphic, and so \( {\alpha }_{\mu } \) is holomorphic. By changing the roles of \( S \) and \( {S}^{\prime } \) we conclude that \( {\alpha }_{\mu } \) is biholomorphic.
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No
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Theorem 5.6. The elements of the modular group \( \operatorname{Mod}\left( S\right) \) are biholomorphic automorphisms of the Teichmüller space \( {T}_{S} \) .
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Proof. By Theorem 5.5, a quasiconformal mapping between the Riemann surfaces \( S \) and \( {S}^{\prime } \) induces a biholomorphic isomorphism \( {T}_{S} \rightarrow {T}_{{S}^{\prime }} \) . The elements of the modular group are such isomorphisms induced by quasi-conformal self-mappings of \( S \) .
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Yes
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Lemma 6.1. Let \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) be a base of \( G \) . Then \( \left( {{\omega }_{1}^{\prime },{\omega }_{2}^{\prime }}\right) \) is a base of \( G \) if and only if\n\n\[{\omega }_{1}^{\prime } = a{\omega }_{1} + b{\omega }_{2},\;{\omega }_{2}^{\prime } = c{\omega }_{1} + d{\omega }_{2},\]\n\n(6.1)\n\nwhere \( a, b, c, d \) are integers and\n\n\[{ad} - {bc} = \pm 1\text{.}\]\n\n(6.2)\n\nIf \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) is normalized, then \( \left( {{\omega }_{1}^{\prime },{\omega }_{2}^{\prime }}\right) \) is normalized if and only if \( {ad} - {bc} = 1 \) .
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Proof. The validity of (6.1) with integral coefficients is clearly a necessary condition. It becomes sufficient if (6.1) can be solved with respect to \( {\omega }_{1} \) and \( {\omega }_{2} \) so that \( {\omega }_{1} \) and \( {\omega }_{2} \) are linear combinations of \( {\omega }_{1}^{\prime } \) and \( {\omega }_{2}^{\prime } \) with coefficients in \( \mathbb{Z} \) . This occurs if and only if (6.2) holds.\n\nFrom (6.1) it follows that\n\n\[ \operatorname{Im}\left( \frac{{\omega }_{1}^{\prime }}{{\omega }_{2}^{\prime }}\right) = \frac{{ad} - {bc}}{{\left| c{\omega }_{1}/{\omega }_{2} + d\right| }^{2}}\operatorname{Im}\left( \frac{{\omega }_{1}}{{\omega }_{2}}\right) .\n\n(6.3)\n\nAssuming that (6.2) is true we see that \( \operatorname{Im}\left( {{\omega }_{1}/{\omega }_{2}}\right) \) and \( \operatorname{Im}\left( {{\omega }_{1}^{\prime }/{\omega }_{2}^{\prime }}\right) \) are simultaneously positive if and only if \( {ad} - {bc} = 1 \) .
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Yes
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Lemma 6.2. Let \( S \) be a torus and \( p \) and \( q \) arbitrary points of \( S \) . Then there is a conformal mapping \( f : S \rightarrow S \) homotopic to the identity such that \( f\left( p\right) = q \) .
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Proof. Let \( \pi : \mathbb{C} \rightarrow S = \mathbb{C}/G \) be the canonical projection and \( z \in {\pi }^{-1}\{ p\}, w \in \) \( {\pi }^{-1}\{ q\} \) . A translation commutes with every \( g \in G \) . Therefore, the mapping \( \zeta \rightarrow \zeta + t\left( {w - z}\right) \) can be projected to a conformal mapping \( {f}_{t} : S \rightarrow S \) for every \( t,0 \leq t \leq 1 \) . As \( t \) varies from 0 to 1, we obtain a homotopy from the identity mapping to \( {f}_{1} = f \), and \( f\left( p\right) = q \) .
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Yes
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Theorem 6.1. Let \( S = \mathbb{C}/G \) and \( {S}^{\prime } = \mathbb{C}/{G}^{\prime } \) be tori and \( \theta : G \rightarrow {G}^{\prime } \) an isomorphism. Then there is a homeomorphism of \( S \) onto \( {S}^{\prime } \) which induces \( \theta \) .
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Proof. Let \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) be a base of \( G \) and suppose that \( \left( {{\omega }_{1},{\omega }_{2}}\right) \rightarrow \left( {{\omega }_{1}^{\prime },{\omega }_{2}^{\prime }}\right) \) under \( \theta \) . Consider the affine transformation \( \alpha \) which fixes 0 and maps \( {\omega }_{i} \) to \( {\omega }_{i}^{\prime }, i = 1,2 \) . Then \( \alpha \) determines \( \theta \), and it projects to a homeomorphism of \( S \) onto \( {S}^{\prime } \) .\n\nLet \( \varphi : S \rightarrow {S}^{\prime } \) be a homeomorphism with a normalized lift \( f \) such that \( f\left( {\omega }_{i}\right) = {\omega }_{i}^{\prime }, i = 1,2 \) . If \( h = f \circ {\alpha }^{-1} \), then \( h\left( {z + {\omega }_{i}^{\prime }}\right) = h\left( z\right) + {\omega }_{i}^{\prime } \) . From this we conclude that \( h \) is sense-preserving (Theorem IV.3.5). Hence \( f \) and \( \alpha \) are simultaneously sense-preserving.\n\nLet \( \tau = {\omega }_{1}/{\omega }_{2},{\tau }^{\prime } = {\omega }_{1}^{\prime }/{\omega }_{2}^{\prime } \) . If \( {\tau }^{\prime } \neq \bar{\tau } \), then\n\n\[ \alpha \left( z\right) = \lambda \left( {z + \mu \bar{z}}\right) \]\n\n(6.4)\n\nDirect calculation shows that\n\n\[ \left| \mu \right| = \left| \frac{{\tau }^{\prime } - \tau }{{\tau }^{\prime } - \bar{\tau }}\right| \]\n\n(6.5)\n\nSuppose that \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) is a normalized base of \( G \), i.e., that \( \operatorname{Im}\tau > 0 \) . By (6.5), we then have \( \operatorname{Im}{\tau }^{\prime } > 0 \) if and only if \( \left| \mu \right| < 1 \) . But \( \left| \mu \right| < 1 \) is equivalent to \( \alpha \) being sense-preserving. It follows that \( \varphi \) is sense-preserving if and only if \( \operatorname{Im}\left( {f\left( {\omega }_{1}\right) /f\left( {\omega }_{2}\right) }\right) > 0 \) . If \( {\tau }^{\prime } = \bar{\tau } \), then \( \varphi \) is sense-reversing and \( \operatorname{Im}\left( {f\left( {\omega }_{1}\right) /}\right. \) \( \left. {f\left( {\omega }_{2}\right) }\right) < 0 \) .\n\nIf \( \left| \mu \right| < 1 \), then (6.4) defines a quasiconformal mapping. Since the projection of a quasiconformal \( \alpha \) is quasiconformal, it follows that all tori are quasiconformally equivalent. We also conclude from the proof of Theorem 6.1, by use of Theorem IV.3.5, that to every sense-preserving homeomorphism \( \varphi \) between two tori \( S \) and \( {S}^{\prime } \) there is a quasiconformal mapping of \( S \) onto \( {S}^{\prime } \) which is homotopic to \( \varphi \) . (This is a new proof of Theorem 1.5 for tori.)
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Yes
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Theorem 6.2. Let \( S = \mathbb{C}/G \) and \( {S}^{\prime } = \mathbb{C}/{G}^{\prime } \) be tori and \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) and \( \left( {{\omega }_{1}^{\prime },{\omega }_{2}^{\prime }}\right) \) normalized bases of \( G \) and \( {G}^{\prime } \) . Then \( S \) and \( {S}^{\prime } \) are conformally equivalent if and only if the points \( {\omega }_{1}/{\omega }_{2} \) and \( {\omega }_{1}^{\prime }/{\omega }_{2}^{\prime } \) are equivalent under the elliptic modular group.
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Proof. We just showed that \( S \) and \( {S}^{\prime } \) are conformally equivalent if and only if there is a \( \lambda \neq 0 \) such that \( \left( {\lambda {\omega }_{1},\lambda {\omega }_{2}}\right) \) is a base of \( {G}^{\prime } \) . From what we said at the end of 6.1 it follows that this is the case if and only if the points \( {\omega }_{1}/{\omega }_{2} \) and \( {\omega }_{1}^{\prime }/{\omega }_{2}^{\prime } \) are equivalent under the elliptic modular group.
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Yes
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Lemma 6.3. Let \( \theta : G \rightarrow {G}^{\prime } \) be an isomorphism generated by a normalized \( K \) - quasiconformal mapping \( f \) . Then\n\n\[{\delta }_{\theta } \leq \frac{1}{2}\log K\]\n\nEquality holds if and only if \( f \) is the affine transformation generating \( \theta \) .
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Proof. Let \( w \) be the affine normalized mapping which generates \( \theta \) . If \( w\left( z\right) = \) \( \lambda \left( {z + \mu \bar{z}}\right) \), we see from (6.5) that \( \left| \mu \right| = \left| {{\tau }^{\prime } - \tau }\right| /\left| {{\tau }^{\prime } - \bar{\tau }}\right| \), where \( \tau = {\omega }_{1}/{\omega }_{2} \) , \( {\tau }^{\prime } = w\left( {\omega }_{1}\right) /w\left( {\omega }_{2}\right) \) . Hence, if \( K \) is the maximal dilatation of \( w \) ,\n\n\[ \log K = \log \frac{1 + \left| \mu \right| }{1 - \left| \mu \right| } = \log \frac{\left| {{\tau }^{\prime } - \bar{\tau }}\right| + \left| {{\tau }^{\prime } - \tau }\right| }{\left| {{\tau }^{\prime } - \bar{\tau }}\right| - \left| {{\tau }^{\prime } - \tau }\right| } = 2{\delta }_{\theta }.\]\n\nConsequently, (6.10) holds as an equality for the affine mapping. Inequality (6.10) and the \
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Yes
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Theorem 6.4. The mapping \( \psi : {T}_{S} \rightarrow H \), defined by\n\n\[ \psi \left( \left\lbrack \varphi \right\rbrack \right) = f\left( {\omega }_{1}\right) /f\left( {\omega }_{2}\right) \]\n\nwhere \( f \) is the normalized lift of \( \varphi \), is a bijective isometry of \( {T}_{S} \) onto the upper half-plane furnished with the hyperbolic metric.
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Proof. The mapping \( \psi \) is injective: If \( \psi \left( {p}_{1}\right) = \psi \left( {p}_{2}\right) \), there are mappings \( {\varphi }_{i} \in {p}_{i}, i = 1,2 \), whose normalized lifts \( {f}_{i} \) satisfy the equations \( {f}_{1}\left( {\omega }_{i}\right) = \lambda {f}_{2}\left( {\omega }_{i}\right) \) , \( i = 1 \) ,2. If \( z \rightarrow {\lambda z} = s\left( z\right) \), then \( {f}_{1} \) and \( s \circ {f}_{2} \) determine the same group isomorphism. It follows that their projections \( {\varphi }_{1} \) and \( \sigma \circ {\varphi }_{2} \) are homotopic. Here \( \sigma \) , as the projection of \( s \), is conformal, and so \( {p}_{1} = {p}_{2} \).\n\nThe mapping \( \psi \) is surjective: Given a point \( z \in H \), we choose an arbitrary non-zero complex number \( {\omega }_{2}^{\prime } \) . After this we set \( {\omega }_{1}^{\prime } = z{\omega }_{2}^{\prime } \) . By Theorem 6.1, there is a homeomorphism \( \varphi \) of \( S \) whose lift \( f \) is normalized and has the properties \( f\left( {\omega }_{i}\right) = {\omega }_{i}^{\prime }, i = 1,2 \) . From the discussion following Theorem 6.1 it follows that \( \varphi \) is sense-preserving. It determines a point \( p \in {T}_{S} \), and we see that \( \psi \left( p\right) = z \).\n\nThe mapping \( \psi \) is an isometry: Given two points \( {p}_{i} \in {T}_{S}, i = 1,2 \), consider their representatives \( {\varphi }_{i} \) with the normalized lifts \( {f}_{i} \) . Let \( \varphi : {\varphi }_{1}\left( S\right) \rightarrow {\varphi }_{2}\left( S\right) \) be the extremal in the class of quasiconformal mappings homotopic to \( {\varphi }_{2} \circ {\varphi }_{1}^{-1} \) and having normalized lifts. For the Teichmüller distance \( {\tau }_{S} \) we then have \( {\tau }_{S}\left( {{p}_{1},{p}_{2}}\right) = \frac{1}{2}\log K \), where \( K \) is the maximal dilatation of \( \varphi \) . On the other hand, if \( f \) is the normalized lift of \( \varphi \), it follows from Lemma 6.3 that \( \frac{1}{2}\log K \) is equal to the hyperbolic distance of the points \( {f}_{1}\left( {\omega }_{1}\right) /{f}_{1}\left( {\omega }_{2}\right) \) and \( f\left( {{f}_{1}\left( {\omega }_{1}\right) }\right) / \) \( f\left( {{f}_{1}\left( {\omega }_{2}\right) }\right) \) . But \( f \circ {f}_{1} \) induces the same group isomorphism as \( {f}_{2} \), and so \( f\left( {{f}_{1}\left( {\omega }_{i}\right) }\right) = {f}_{2}\left( {\omega }_{i}\right), i = 1,2 \) . Thus the Teichmüller distance \( {\tau }_{S}\left( {{p}_{1},{p}_{2}}\right) \) coincides with the hyperbolic distance between \( \psi \left( {p}_{1}\right) \) and \( \psi \left( {p}_{2}\right) \), and the theorem is proved.
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Yes
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Theorem 6.5. The mapping\n\n\[ \left\lbrack z\right\rbrack \rightarrow z \]\n\n(6.13)\n\nis a bijective isometry of the Teichmüller space \( {T}_{S} \) onto the hyperbolic unit disc D.
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Proof. The theorem follows immediately from the fact that (6.12) is a bijective isometry of \( {T}_{S} \) onto the hyperbolic upper half-plane.
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Yes
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